Displaying 1-10 of 33 results found.
Numerators of Cauchy numbers of first type.
(Formerly M5067)
+10
107
1, 1, -1, 1, -19, 9, -863, 1375, -33953, 57281, -3250433, 1891755, -13695779093, 24466579093, -132282840127, 240208245823, -111956703448001, 4573423873125, -30342376302478019, 56310194579604163
COMMENTS
The corresponding denominators are given in A006233. -a(n+1), n>=0, also numerators from e.g.f. 1/x-1/log(1+x), with denominators A075178(n). |a(n+1)|, n>=0, numerators from e.g.f. 1/x+1/log(1-x) with denominators A075178(n). For formula of unsigned a(n) see A075178. The signed rationals a(n)/ A006233(n) provide the a-sequence for the Stirling2 Sheffer matrix A048993. See the W. Lang link concerning Sheffer a- and z-sequences. Cauchy numbers of the first type are also called Bernoulli numbers of the second kind.
Named after the French mathematician, engineer and physicist Augustin-Louis Cauchy (1789-1857). - Amiram Eldar, Jun 17 2021
REFERENCES
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
Harold Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics, Cambridge, 1946, p. 259.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Donatella Merlini, Renzo Sprugnoli and M. Cecilia Verri, The Cauchy numbers, Discrete Math., Vol. 306, No. 16 (2006), pp. 1906-1920.
FORMULA
Numerator of integral of x(x-1)...(x-n+1) from 0 to 1.
E.g.f.: x/log(1+x). (Note: the numerator of the coefficient of x^n/n! is a(n) - Michael Somos, Jul 12 2014) Sum_{k=1..n} 1/k = C + log(n) + 1/(2n) + Sum_{k=2..inf} |a(n)|/ A075178(n-1) * 1/(n*(n+1)*...*(n+k-1)) (section 0.131 in Gradshteyn and Ryzhik tables). - Ralf Stephan, Jul 12 2014 a(n) = numerator(f(n) * n!), where f(0) = 1, f(n) = Sum_{k=0..n-1} (-1)^(n-k+1) * f(k) / (n-k+1). - Daniel Suteu, Feb 23 2018 Sum_{k = 1..n} (1/k) = A001620 + log(n) + 1/(2n) - Sum_{k >= 2} abs((a(k)/ A006233(k)/k/(Product_{j = 0..k-1} (n-j)))), (see I. S. Gradsteyn, I. M. Ryzhik). - A.H.M. Smeets, Nov 14 2018
EXAMPLE
1, 1/2, -1/6, 1/4, -19/30, 9/4, -863/84, 1375/24, -33953/90, ...
MAPLE
seq(numer(add(stirling1(n, k)/(k+1), k=0..n)), n=0..20); # Peter Luschny, Apr 28 2009
MATHEMATICA
a[n_] := Numerator[ Sum[ StirlingS1[n, k]/(k + 1), {k, 0, n}]]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Nov 03 2011, after Maple *) a[n_] := Numerator[ Integrate[ Gamma[x+1]/Gamma[x-n+1], {x, 0, 1}]]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jul 29 2013 *) a[ n_] := If[ n < 0, 0, (-1)^n Numerator @ Integrate[ Pochhammer[ -x, n], {x, 0, 1}]]; (* Michael Somos, Jul 12 2014 *) a[ n_] := If[ n < 0, 0, Numerator [ n! SeriesCoefficient[ x / Log[ 1 + x], {x, 0, n}]]]; (* Michael Somos, Jul 12 2014 *) Join[{1}, Array[Numerator[(1/#) Integrate[Product[(x - k), {k, 0, # - 1}], {x, 0, 1}]] &, 25]] (* Michael De Vlieger, Nov 13 2018 *)
PROG
(Sage)
f, R, C = 1, [1], [1]+[0]*(len-1)
for n in (1..len-1):
for k in range(n, 0, -1):
C[k] = -C[k-1] * k / (k + 1)
C[0] = -sum(C[k] for k in (1..n))
R.append((C[0]*f).numerator())
f *= n
return R
(PARI) for(n=0, 20, print1(numerator( sum(k=0, n, stirling(n, k, 1)/(k+1)) ), ", ")) \\ G. C. Greubel, Nov 13 2018 (Magma) [Numerator((&+[StirlingFirst(n, k)/(k+1): k in [0..n]])): n in [0..20]]; // G. C. Greubel, Nov 13 2018 (Python) # Results are abs values
from fractions import gcd
aa, n, sden = [0, 1], 1, 1
while n < 20:
j, snom, sden, a = 1, 0, (n+1)*sden, 0
while j < len(aa):
snom, j = snom+aa[j]*(sden//(j+1)), j+1
nom, den = snom, sden
print(n, nom//gcd(nom, den))
aa, j = aa+[-aa[j-1]], j-1
while j > 0:
aa[j], j = n*aa[j]-aa[j-1], j-1
(Python)
from fractions import Fraction
from sympy.functions.combinatorial.numbers import stirling
def A006232(n): return sum(Fraction(stirling(n, k, kind=1, signed=True), k+1) for k in range(n+1)).numerator # Chai Wah Wu, Jul 09 2023
Denominators of Cauchy numbers of first type.
(Formerly M1558)
+10
37
1, 2, 6, 4, 30, 4, 84, 24, 90, 20, 132, 8, 5460, 840, 360, 48, 1530, 4, 1596, 168, 1980, 1320, 8280, 80, 81900, 6552, 1512, 112, 3480, 80, 114576, 7392, 117810, 7140, 1260, 8, 3838380, 5928, 936, 48, 81180, 440, 1191960, 55440, 869400, 38640, 236880, 224
COMMENTS
The corresponding numerators are given in A006232. The signed rationals A006232(n)/a(n) provide the a-sequence for the Stirling2 Sheffer matrix A048993. See the W. Lang link concerning Sheffer a- and z-sequences. Cauchy numbers of the first type are also called Bernoulli numbers of the second kind.
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
H. Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics, Cambridge, 1946, p. 259.
L. Jolley, Summation of Series, Chapman and Hall, London, 1925, pp. 14-15 (formula 70).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Donatella Merlini, Renzo Sprugnoli and M. Cecilia Verri, The Cauchy numbers, Discrete Math. 306 (2006), no. 16, 1906-1920.
FORMULA
Denominator of integral of x(x-1)...(x-n+1) from 0 to 1.
E.g.f.: x/log(1+x).
a(n) = denominator(f(n) * n!), where f(0) = 1, f(n) = Sum_{k=0..n-1} (-1)^(n-k+1) * f(k) / (n-k+1). - Daniel Suteu, Feb 23 2018 Sum_{k = 1..n} (1/k) = A001620 + log(n) + 1/(2*n) - Sum_{k >= 2} abs(( A006232(k)/a(k)/k/(Product_{j = 0..k-1} (n-j)))), (see I. S. Gradsteyn, I. M. Ryzhik). - A.H.M. Smeets, Nov 14 2018
EXAMPLE
1, 1/2, -1/6, 1/4, -19/30, 9/4, -863/84, 1375/24, -33953/90,...
MAPLE
seq(denom(add(stirling1(n, k)/(k+1), k=0..n)), n=0..12); # Peter Luschny, Apr 28 2009
MATHEMATICA
With[{nn=50}, Denominator[CoefficientList[Series[x/Log[1+x], {x, 0, nn}], x] Range[0, nn]!]] (* Harvey P. Dale, Oct 28 2011 *) Join[{1}, Array[Abs@Denominator[ Integrate[Product[(x - k), {k, 0, # - 1}], {x, 0, 1}]] &, 50]] (* Michael De Vlieger, Nov 13 2018 *)
PROG
(PARI) for(n=0, 50, print1(denominator( sum(k=0, n, stirling(n, k, 1)/(k+1)) ), ", ")) \\ G. C. Greubel, Nov 13 2018 (Magma) [Denominator((&+[StirlingFirst(n, k)/(k+1): k in [0..n]])): n in [0..50]]; // G. C. Greubel, Nov 13 2018 (Sage)
f, R, C = 1, [1], [1]+[0]*(len-1)
for n in (1..len-1):
for k in range(n, 0, -1):
C[k] = -C[k-1] * k / (k + 1)
C[0] = -sum(C[k] for k in (1..n))
R.append((C[0]*f).denominator())
f *= n+1
return R
(Python) # Results are abs values
from fractions import gcd
aa, n, sden = [0, 1], 1, 1
print(0, 1)
while n < 20:
j, snom, sden, a = 1, 0, (n+1)*sden, 0
while j < len(aa):
snom, j = snom+aa[j]*(sden//(j+1)), j+1
nom, den = snom, sden
print(n, den//gcd(nom, den))
aa, j = aa+[-aa[j-1]], j-1
while j > 0:
aa[j], j = n*aa[j]-aa[j-1], j-1
(Python)
from fractions import Fraction
from sympy.functions.combinatorial.numbers import stirling
def A006233(n): return sum(Fraction(stirling(n, k, kind=1, signed=True), k+1) for k in range(n+1)).denominator # Chai Wah Wu, Jul 09 2023
KEYWORD
nonn,frac,nice,easy,changed
Numerators of Cauchy numbers of second type (= Bernoulli numbers B_n^{(n)}).
(Formerly M3790 N1545)
+10
27
1, 1, 5, 9, 251, 475, 19087, 36799, 1070017, 2082753, 134211265, 262747265, 703604254357, 1382741929621, 8164168737599, 5362709743125, 8092989203533249, 15980174332775873, 12600467236042756559
COMMENTS
These coefficients (with alternating signs) are also known as the Nørlund [or Norlund, Noerlund or Nörlund] numbers. [After the Danish mathematician Niels Erik Nørlund (1885-1981). - Amiram Eldar, Jun 17 2021] The denominators are found in A002790. The alternating rational sequence ((-1)^n)*a(n)/ A002790(n)is the z-sequence for the Stirling2 triangle A008277(n+1,k+1), n>=k>=0. This is the Sheffer (exp(x),exp(x)-1) triangle. See the W. Lang link under A006232 for Sheffer a- and z-sequences with references, and the conversion to S. Roman's notation. The a-sequence is A006232(n)/ A006233(n). - Wolfdieter Lang, Oct 06 2011 [This is the Sheffer triangle A007318* A048993. Added Jun 20 2017] A simple series with the signless Cauchy numbers of second type C2(n) leads to Euler's constant: gamma = 1 - Sum_{n >=1} C2(n)/(n*(n+1)!) = 1 - 1/4 - 5/72 - 1/32 - 251/14400 - 19/1728 - 19087/2540160 - ..., see references [Blagouchine] below, as well as A075266 and A262235. - Iaroslav V. Blagouchine, Sep 15 2015
REFERENCES
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969.
Louis Melville Milne-Thompson, Calculus of Finite Differences, 1951, p. 136.
N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Ibrahim M. Alabdulmohsin, The Language of Finite Differences, in Summability Calculus: A Comprehensive Theory of Fractional Finite Sums, Springer, Cham, 2018, pp. 133-149. Donatella Merlini, Renzo Sprugnoli and M. Cecilia Verri, The Cauchy numbers, Discrete Math., Vol. 306, No. 16 (2006), pp. 1906-1920.
FORMULA
Numerator of integral of x(x+1)...(x+n-1) from 0 to 1.
a(n) = numerator(n!*v(n)), where v(n) = 1 - Sum_{i=0..n-1} v(i)/(n-i+1), v(0)=1. - Vladimir Kruchinin, Aug 28 2013
EXAMPLE
1, 1/2, 5/6, 9/4, 251/30, 475/12, 19087/84, 36799/24, 1070017/90, ...
MAPLE
seq(numer(add((-1)^(n-k)*Stirling1(n, k)/(k+1), k=0..n)), n=0..10); # Peter Luschny, Apr 28 2009
MATHEMATICA
a[ n_] := If[ n < 0, 0, (-1)^n Numerator @ NorlundB[ n, n]]; (* Michael Somos, Jul 12 2014 *) a[ n_] := If[ n < 0, 0, Numerator@Integrate[ Pochhammer[ x, n], {x, 0, 1}]]; (* Michael Somos, Jul 12 2014 *) a[ n_] := If[ n < 0, 0, Numerator[ n! SeriesCoefficient[ -x / ((1 - x) Log[ 1 - x]), {x, 0, n}]]]; (* Michael Somos, Jul 12 2014 *) a[ n_] := If[ n < 0, 0, (-1)^n Numerator[ n! SeriesCoefficient[ (x / (Exp[x] - 1))^n, {x, 0, n}]]]; (* Michael Somos, Jul 12 2014 *)
PROG
(Maxima) v(n):=if n=0 then 1 else 1-sum(v(i)/(n-i+1), i, 0, n-1);
(Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(-x/((1-x)*Log(1-x)) )); [Numerator(Factorial(n-1)*b[n]): n in [1..m-1]]; // G. C. Greubel, Oct 29 2018
Denominators of logarithmic numbers (also of Gregory coefficients G(n)).
(Formerly M2017 N0797)
+10
25
1, 2, 12, 24, 720, 160, 60480, 24192, 3628800, 1036800, 479001600, 788480, 2615348736000, 475517952000, 31384184832000, 689762304000, 32011868528640000, 15613165568, 786014494949376000, 109285437800448000
COMMENTS
Denominator of the determinant of the (n+1) X (n+1) matrix with 1's along the superdiagonal, (1/2)'s along the main diagonal, (1/3)'s along the subdiagonal, etc., and 0's everywhere else. - John M. Campbell, Dec 01 2011
REFERENCES
E. Isaacson and H. Bishop, Analysis of Numerical Methods, ISBN 0 471 42865 5, 1966, John Wiley and Sons, pp. 318-319. - Rudi Huysmans (rudi_huysmans(AT)hotmail.com), Apr 10 2000
Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 266.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
1/log(1+x) = Sum_{n>=-1} ( A002206(n)/a(n)) * x^n. A002206(n)/ A002207(n) = (1/n!) * Sum_{j=1..n+1} Bernoulli(j)/j * S_1(n, j-1), where S_1(n,k) is the Stirling number of the first kind. - Barbara Margolius (b.margolius(AT)csuohio.edu), Jan 21 2002
G(0) = 0, G(n) = Sum_{i=1..n} (-1)^(i+1)*G(n-i)/(i+1) + (-1)^(n+1)*n/(2*(n+1)*(n+2)).
G(n) = (1/(n+1)!)*Integral_{x=0..1} x*(x-n)_n dx, where (a)_n is the Pochhammer symbol. - Vladimir Reshetnikov, Oct 22 2015 a(n) = denominator(f(n+1)), where f(0) = 1, f(n) = Sum_{k=0..n-1} (-1)^(n-k+1) * f(k) / (n-k+1). - Daniel Suteu, Nov 15 2018
EXAMPLE
Logarithmic numbers are 1, 1/2, -1/12, 1/24, -19/720, 3/160, -863/60480, 275/24192, -33953/3628800, 8183/1036800, -3250433/479001600, 4671/788480, -13695779093/2615348736000, 2224234463/475517952000, ... = A002206/ A002207
MAPLE
series(1/log(1+x), x, 25);
with(combinat, stirling1):seq(denom(1/i!*sum(bernoulli(j)/(j)*stirling1(i, j-1), j=1..i+1)), i=1..24);
MATHEMATICA
Table[Denominator[Det[Array[Sum[KroneckerDelta[#1, #2+q]*1/(q+2)^1, {q, -1, n+1}] &, {n+1, n+1}]]], {n, 0, 20}] (* John M. Campbell, Dec 01 2011 *) a[n_] := Denominator[n!^-1*Sum[BernoulliB[j]/j*StirlingS1[n, j-1], {j, 1, n+1}]]; a[-1] = 1; Table[a[n], {n, -1, 18}] (* Jean-François Alcover, May 16 2012, after Maple *) Denominator@Table[Integrate[x Pochhammer[x - n, n], {x, 0, 1}]/(n + 1)!, {n, -1, 20}] (* Vladimir Reshetnikov, Oct 22 2015 *) Denominator@CoefficientList[x/Log[1+x] + O[x]^20, x] (* Oliver Seipel, Jul 06 2024 *)
PROG
(PARI) a(n) = denominator(sum(k=0, n+1, stirling(n+1, k, 1)/((n+1)!*(k+1)))); \\ Michel Marcus, Mar 20 2018 (Python)
from math import factorial
from fractions import Fraction
from sympy.functions.combinatorial.numbers import stirling
def A002207(n): return (sum(Fraction(stirling(n+1, k, kind=1, signed=True), k+1) for k in range(n+2))/factorial(n+1)).denominator # Chai Wah Wu, Feb 12 2023 (SageMath)
from functools import cache
@cache
def h(n):
return (-sum((-1)**k * h(n - k) / (k + 1) for k in range(1, n + 1))
+ (-1)**n * n / (2*(n + 1)*(n + 2)))
def a(n): return h(n).denom() if n > 0 else n + 2
print([a(n) for n in range(-1, 19)]) # Peter Luschny, Dec 12 2023
Denominators of Cauchy numbers of second type (= Bernoulli numbers B_n^{(n)}).
(Formerly M1559 N0608)
+10
23
1, 2, 6, 4, 30, 12, 84, 24, 90, 20, 132, 24, 5460, 840, 360, 16, 1530, 180, 7980, 840, 13860, 440, 1656, 720, 81900, 6552, 216, 112, 3480, 240, 114576, 7392, 117810, 2380, 1260, 72, 3838380, 207480, 32760, 560, 568260, 27720, 238392, 55440, 869400, 2576, 236880
COMMENTS
The numerators are given in A002657. These coefficients (with alternating signs) are also known as the Nørlund [or Norlund, Noerlund or Nörlund] numbers.
A simple series with the signless Cauchy numbers of second type C2(n) leads to Euler's constant: gamma = 1 - Sum_{n >=1} C2(n)/(n*(n+1)!) = 1 - 1/4 - 5/72 - 1/32 - 251/14400 - 19/1728 - 19087/2540160 - ..., see references [Blagouchine] below, as well as A075266 and A262235. - Iaroslav V. Blagouchine, Sep 15 2015
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 136.
N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Donatella Merlini, Renzo Sprugnoli and M. Cecilia Verri, The Cauchy numbers, Discrete Math. 306 (2006), no. 16, 1906-1920.
FORMULA
Denominator of integral of x(x+1)...(x+n-1) from 0 to 1.
a(n) = denominator(n!*v(n)), where v(n) = 1 - Sum_{i=0..n-1} v(i)/(n-i+1), v(0)=1. - Vladimir Kruchinin, Aug 28 2013
EXAMPLE
1, 1/2, 5/6, 9/4, 251/30, 475/12, 19087/84, 36799/24, 1070017/90, ...
MAPLE
denom(add((-1)^k*stirling1(n, k)/(k+1), k=0..n)) ;
PROG
(Maxima)
v(n):=if n=0 then 1 else 1-sum(v(i)/(n-i+1), i, 0, n-1);
(Magma) m:=60; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(-x/((1-x)*Log(1-x)) )); [Denominator(Factorial(n-1)*b[n]): n in [1..m-1]]; // G. C. Greubel, Oct 28 2018
Numerators of coefficients for numerical integration.
(Formerly M3737 N1527)
+10
22
1, 1, 5, 3, 251, 95, 19087, 5257, 1070017, 25713, 26842253, 4777223, 703604254357, 106364763817, 1166309819657, 25221445, 8092989203533249, 85455477715379, 12600467236042756559, 1311546499957236437, 8136836498467582599787
COMMENTS
The denominators are given in A002209. a(n) is (-1)^n times the numerator of the "reverse" multiple zeta value zeta_n^R(0,0,...,0) for n > 0. - Jonathan Sondow, Nov 29 2006 The denominators are given in A002209.
REFERENCES
E. Isaacson and H. B. Keller, Analysis of Numerical Methods, ISBN 0 471 42865 5, 1966, John Wiley and Sons, pp. 318-319.
Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 529.
N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
G.f. of rationals a(n)/ A002209(n): -x/((1-x)*log(1-x)). Let K_i = a(i)/ A002209(i), for i >= 1, and [i n] = Stirling numbers of the first kind ( A048994), {i n} = Stirling numbers of the second kind ( A048993) and B_i the original Bernoulli numbers ( A164555/ A027642). Then K_i = ((-1)^(i-1) / (i-1)!)*Sum_{n=1..i} [i n]*B_n/n and B_i = i*Sum_{n=1..i} (-1)^(n-1)*{i n}*(n-1)!*K_n. - Rudi Huysmans, rudi_huysmans(AT)hotmail.com [see the second Mathematica program for K_n = a[n_] with B_k = (-1)^k * BernoulliB[k]. - Wolfdieter Lang, Aug 09 2017] a(n) = numerator((-1)^n*Sum_{k=0..n} (k!*Stirling2(n,k)* Stirling1(n+k,n))/(n+k)!). - Vladimir Kruchinin, Feb 02 2013 a(n) = numerator(v(n)), where v(n) = 1 - Sum_{i=0..n-1} v(i)/(n-i+1), v(0)=1. - Vladimir Kruchinin, Aug 28 2013 a(n) = numerator((1/(n-1)!)*Sum_{k=0..n} ((-1)^(n-k)*binomial(2*n,n-k)*Stirling2(n+k,k))/(n+k)), n > 0, a(0)=1. - Vladimir Kruchinin, Apr 05 2016 a(n) = numerator(((-1)^n/n!)*Sum_{k=0..n} Stirling1(n+1,k+1)/(k+1)). - Vladimir Kruchinin, Oct 12 2016
EXAMPLE
1, 1/2, 5/12, 3/8, 251/720, 95/288, 19087/60480, 5257/17280, 1070017/3628800, 25713/89600, 26842253/95800320, 4777223/17418240, 703604254357/2615348736000, 106364763817/402361344000, ... = A002208/ A002209.
MAPLE
r := proc(n) option remember; if n=0 then 1 else 1 - add(r(k)/(n-k+1), k=0..n-1) fi end: seq(numer(r(n)), n=0..20); # Peter Luschny, Feb 16 2020
MATHEMATICA
Numerator/@CoefficientList[Series[-x/((1-x)Log[1-x]), {x, 0, 20}], x] (* Harvey P. Dale, May 04 2011 *) a[0] = 1; a[n_] := (-1)^n*Sum[(-1)^(k+1)*BernoulliB[k]*StirlingS1[n, k]/k, {k, 1, n}]/(n-1)!; Table[a[n], {n, 0, 20}] // Numerator (* Jean-François Alcover, Sep 27 2012, after Rudi Huysmans's formula *)
PROG
(Maxima)
a(n):=if n=0 then 1 else 1/(n-1)!*sum(((-1)^(n-k)*binomial(2*n, n-k)*stirling2(n+k, k))/(n+k), k, 0, n); /* Vladimir Kruchinin, Apr 05 2016 */ a(n):=num(((-1)^(n)*sum(stirling1(n+1, k+1)/(k+1), k, 0, n))/(n)!); /* Vladimir Kruchinin, Oct 12 2016 */ (Python)
from math import factorial
from fractions import Fraction
from sympy.functions.combinatorial.numbers import stirling
def A002208(n): return (-1 if n&1 else 1)*(sum(Fraction(stirling(n+1, k+1, kind=1, signed=True), k+1) for k in range(n+1))/factorial(n)).numerator # Chai Wah Wu, Jul 09 2023
Denominators of coefficients for numerical integration.
(Formerly M2015 N0796)
+10
22
1, 2, 12, 8, 720, 288, 60480, 17280, 3628800, 89600, 95800320, 17418240, 2615348736000, 402361344000, 4483454976000, 98402304, 32011868528640000, 342372925440000, 51090942171709440000, 5377993912811520000, 33720021833328230400000
COMMENTS
a(n) is the denominator of the "reverse" multiple zeta value zeta_n^R(0,0,...,0) for n > 0. - Jonathan Sondow, Nov 29 2006 The numerators are given in A002208.
REFERENCES
Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 529.
N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
G.f. of A002208(n)/a(n): -x/((1-x)*log(1-x)). a(n) = denominator(v(n)), where v(n) = 1 - Sum_{i=0..n-1} v(i)/(n-i+1), v(0)=1. - Vladimir Kruchinin, Aug 28 2013 a(n) = denominator(((-1)^n/n!)*Sum_{k=0..n} Stirling1(n+1,k+1)/(k+1)). - Vladimir Kruchinin, Oct 12 2016
EXAMPLE
1, 1/2, 5/12, 3/8, 251/720, 95/288, 19087/60480, 5257/17280, 1070017/3628800, 25713/89600, 26842253/95800320, 4777223/17418240, 703604254357/2615348736000, 106364763817/402361344000, ... = A002208/ A002209.
MATHEMATICA
a[0] = 1; a[n_] := (-1)^n*Sum[(-1)^(k+1)*BernoulliB[k]*StirlingS1[n, k]/k, {k, 1, n}]/(n-1)!; Table[a[n], {n, 0, 20}] // Denominator (* Jean-François Alcover, Sep 27 2012, after Rudi Huysmans's formula for A002208 *) Denominator[CoefficientList[Series[-x/((1-x)Log[1-x]), {x, 0, 20}], x]] (* Harvey P. Dale, Feb 01 2013 *)
PROG
(Maxima)
a(n):=denom(((-1)^(n)*sum(stirling1(n+1, k+1)/(k+1), k, 0, n))/(n)!); /* Vladimir Kruchinin, Oct 12 2016 */ (Python)
from math import factorial
from fractions import Fraction
from sympy.functions.combinatorial.numbers import stirling
def A002209(n): return (sum(Fraction(stirling(n+1, k+1, kind=1, signed=True), k+1) for k in range(n+1))/factorial(n)).denominator # Chai Wah Wu, Jul 09 2023
Denominators of a sequence leading to gamma = A001620.
+10
14
2, 24, 72, 2880, 800, 362880, 169344, 29030400, 9331200, 4790016000, 8673280, 31384184832000, 6181733376000, 439378587648000, 10346434560000, 512189896458240000, 265423814656, 14148260909088768000, 2076423318208512000, 96342919523794944000000, 74538995631567667200000
COMMENTS
gamma = 1/2 + 1/24 + 1/72 + 19/2880 + 3/800 + 863/362880 + 275/169344 + ... = ( A002206 unsigned=reduced A141417(n+1)/ A091137(n+1))/a(n) is an old formula based on Gregory's A002206/ A002207. This formula for Euler's constant was discovered circa 1780-1790 by the Italian mathematicians Gregorio Fontana (1735-1803) and Lorenzo Mascheroni (1750-1800), and was subsequently rediscovered several times (in particular, by Ernst Schröder in 1879, Niels E. Nørlund in 1923, Jan C. Kluyver in 1924, Charles Jordan in 1929, Kenter in 1999, and Victor Kowalenko in 2008). For more details, see references below. - Iaroslav V. Blagouchine, May 03 2015
EXAMPLE
a(0)=1*2, a(1)=2*12, a(2)=3*24, a(3)=4*720.
MATHEMATICA
g[n_]:=Sum[StirlingS1[n, l]/(l+1), {l, 1, n}]/(n*n!); a[n_]:=Denominator[g[n]]; Table[a[n], {n, 1, 30}] (* Iaroslav V. Blagouchine, May 03 2015 *) g[n_] := Sum[ BernoulliB[j]/j * StirlingS1[n, j-1], {j, 1, n+1}] / n! ; a[n_] := (n+1)*Denominator[g[n]]; Table[a[n], {n, 0, 20}]
(* or *) max = 20; Denominator[ CoefficientList[ Series[ 1/Log[1 + x] - 1/x, {x, 0, max}], x]]*Range[max+1] (* Jean-François Alcover, Sep 04 2013 *)
Decimal expansion of log_2 e.
(Formerly M3221)
+10
8
1, 4, 4, 2, 6, 9, 5, 0, 4, 0, 8, 8, 8, 9, 6, 3, 4, 0, 7, 3, 5, 9, 9, 2, 4, 6, 8, 1, 0, 0, 1, 8, 9, 2, 1, 3, 7, 4, 2, 6, 6, 4, 5, 9, 5, 4, 1, 5, 2, 9, 8, 5, 9, 3, 4, 1, 3, 5, 4, 4, 9, 4, 0, 6, 9, 3, 1, 1, 0, 9, 2, 1, 9, 1, 8, 1, 1, 8, 5, 0, 7, 9, 8, 8, 5, 5, 2, 6, 6, 2, 2, 8, 9, 3, 5, 0, 6, 3, 4, 4, 4, 9, 6, 9, 9
COMMENTS
Around 1670, James Gregory discovered by inversion of 1 - 1/2 + 1/3 - 1/4 + 1/5 - ... = log(2) that 1 + 1/2 - 1/12 + 1/24 - 19/720 + (27/1440 = 3/160) - 863/60480 + ... = 1/log(2). See formula with A002206 and A002207. See also A141417 signed / A091137; case i = 0 in A165313. First row in array p. 36 of the reference. - Paul Curtz, Sep 12 2011 This constant 1/log(2) is also related to the asymptotic evaluation of the maximum number of subtraction steps required to compute gcd(m, n) by the binary Euclidean algorithm, m and n being odd and chosen at random. - Jean-François Alcover, Jun 23 2014, after Steven Finch
REFERENCES
Paul Curtz, Intégration numérique des systèmes différentiels .. , note n° 12, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.18 Porter-Hensley constants, p. 159.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 25, equation 25:14:3 at page 232.
FORMULA
1/log(2) = Sum_{n = -infinity..infinity} (2^n / (1 + 2^2^n)). - Nicolas Nagel, Mar 16 2018 More generally: 1/log(2) = Sum_{n = -infinity..infinity} (2^(n+x) / (1 + 2^2^(n+x))) for all real x. - Nicolas Nagel, Jul 02 2019 Equals 1 + Sum_{k>=1} 1/(2^k * (1 + 2^(1/2^k))).
Equals Product_{k>=1} ((1 + 2^(1/2^k))/2). (End)
EXAMPLE
1.442695040888963407359924681...
Numerators of a semi-convergent series leading to the first Stieltjes constant gamma_1.
+10
8
-1, 11, -137, 121, -7129, 57844301, -1145993, 4325053069, -1848652896341, 48069674759189, -1464950131199, 105020512675255609, -22404210159235777, 1060366791013567384441, -15899753637685210768473787, 2241672100026760127622163469, -8138835628210212414423299
COMMENTS
gamma_1 = - 1/12 + 11/720 - 137/15120 + 121/11200 - 7129/332640 + 57844301/908107200 - ..., see formulas (46)-(47) in the reference below.
FORMULA
a(n) = numerator(-B_{2n}*H_{2n-1}/(2n)), where B_n and H_n are Bernoulli and harmonic numbers respectively.
a(n) = numerator(Zeta(1 - 2*n)*(Psi(2*n) + gamma)), where gamma is Euler's gamma. - Peter Luschny, Apr 19 2018
EXAMPLE
Numerators of -1/12, 11/720, -137/15120, 121/11200, -7129/332640, 57844301/908107200, ...
MAPLE
a := n -> numer(Zeta(1 - 2*n)*(Psi(2*n) + gamma)):
MATHEMATICA
a[n_] := Numerator[-BernoulliB[2*n]*HarmonicNumber[2*n - 1]/(2*n)]; Table[a[n], {n, 1, 20}]
PROG
(PARI) a(n) = numerator(-bernfrac(2*n)*sum(k=1, 2*n-1, 1/k)/(2*n)); \\ Michel Marcus, Sep 23 2015
CROSSREFS
Cf. A001620, A002206, A195189, A075266, A262235, A001067, A006953, A082633, A262383 (denominators of this series), A086279, A086280, A262387.
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