Displaying 1-10 of 13 results found.
Numerators of coefficients of log(1+x)/sqrt(1+x).
(Formerly M5128 N2223)
+0
2
1, 1, 23, 11, 563, 1627, 88069, 1423, 1593269, 7759469, 31730711, 46522243, 3788707301, 2888008157, 340028535787, 41743955887, 10823198495797, 2738032559863, 409741429887649, 25876414060339, 17141894231615609
REFERENCES
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).
PROG
(PARI) x='x+O('x^66); abs(apply(t->numerator(t), Vec(log(1+x)/sqrt(1+x)))) \\ Joerg Arndt, Apr 18 2014
Coefficients for numerical differentiation.
(Formerly M5166 N2243)
+0
0
1, 24, 640, 7168, 294912, 2883584, 54525952, 167772160, 36507222016, 326417514496, 5772436045824, 50577534877696, 1759218604441600, 15199648742375424, 261208778387488768, 2233785415175766016, 101457092405402533888
REFERENCES
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
a(n) is the denominator of(-1)^(n-1)*Cn-1{1^2..(2n-1)^2}/((2n+1)!*2^(2n)), where Cn{1^2..(2n+1)^2} is equal to 1 when n=0, otherwise, it is the sum of the products of all possible combinations, of size n, of the numbers (2k+1)^2 with k=0,1,..,n. - Sean A. Irvine, after Ruperto Corso, Mar 29 2014
MAPLE
with(combinat): a:=n->add(mul(k, k=j), j=choose([seq((2*i-1)^2, i=1..n)], n))*(-1)^(n-1)/(2^(2*n)*(2*n+1)!):seq(a(n), n=0..20); # Sean A. Irvine, after Ruperto Corso
Numerator of Sum_{i+j+k=n; i,j,k > 0} 1/(i*j*k).
(Formerly M2651 N1058)
+0
4
1, 3, 7, 15, 29, 469, 29531, 1303, 16103, 190553, 128977, 9061, 30946717, 39646461, 58433327, 344499373, 784809203, 169704792667, 665690574539, 5667696059, 337284946763, 7964656853269, 46951444927823, 284451446729, 1597747168263479, 816088653136373
COMMENTS
Numerators of coefficients for numerical differentiation.
For prime p >= 5, a(p) == -2*Bernoulli(p-3) (mod p). (See Zhao link.) - Michel Marcus, Feb 05 2016
REFERENCES
W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables).
A. N. Lowan, H. E. Salzer and A. Hillman, A table of coefficients for numerical differentiation, Bull. Amer. Math. Soc., 48 (1942), 920-924.
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.: (-log(1-x))^3 (for fractions A002545(n)/ A002546(n)). - Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002 A002545(n)/ A002546(n) = 6*Stirling_1(n+3, 3)*(-1)^n/(n+3)!. - Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002
MAPLE
seq(numer(-Stirling1(j, 3)/j!*3!*(-1)^j), j=3..50); # Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002
MATHEMATICA
Table[Sum[1/i/j/(n-i-j), {i, n-2}, {j, n-i-1}], {n, 3, 100}] (* Ryan Propper *)
EXTENSIONS
More terms from Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002
Denominator of Sum_{i+j+k=n; i,j,k > 0} 1/(i*j*k).
(Formerly M1110 N0424)
+0
2
1, 2, 4, 8, 15, 240, 15120, 672, 8400, 100800, 69300, 4950, 17199000, 22422400, 33633600, 201801600, 467812800, 102918816000, 410646075840, 3555377280, 215100325440, 5162407810560, 30920671782000, 190281057120, 1085315579548200, 562756226432400, 22969641895200
COMMENTS
Denominators of coefficients for numerical differentiation.
REFERENCES
W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables).
A. N. Lowan, H. E. Salzer and A. Hillman, A table of coefficients for numerical differentiation, Bull. Amer. Math. Soc., 48 (1942), 920-924.
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.: (-log(1-x))^3 (for fractions A002545(n)/ A002546(n)). - Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002 A002545(n)/ A002546(n) = 6*Stirling_1(n+3, 3)(-1)^n/(n+3)!. - Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002
MAPLE
seq(denom(-Stirling1(j, 3)/j!*3!*(-1)^j), j=3..50); # Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002
MATHEMATICA
Denominator[Table[Sum[1/i/j/(n-i-j), {i, n-2}, {j, n-i-1}], {n, 3, 100}]] (* Ryan Propper *)
EXTENSIONS
More terms from Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002
Denominators of coefficients for numerical differentiation.
(Formerly M5177 N2249)
+0
3
1, 24, 5760, 322560, 51609600, 13624934400, 19837904486400, 2116043145216, 20720294477955072, 15747423803245854720, 131978409017679544320, 72852081777759108464640, 151532330097738945606451200, 2828603495157793651320422400, 19687080326298243813190139904000
REFERENCES
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
a(n) is the denominator of (-1)^(n-1)*Cn-1{1^2..(2n-1)^2}/((2n)!*2^(2n-3)), where Cn{1^2..(2n+1)^2} is equal to 1 when n=0, otherwise, it is the sum of the products of all possible combinations, of size n, of the numbers (2k+1)^2 with k=0,1,..,n. - Ruperto Corso, Dec 15 2011
MAPLE
with(combinat): a:=n->add(mul(k, k=j), j=choose([seq((2*i-1)^2, i=1..n)], n-1))*(-1)^(n-1)/(2^(2*n-3)*(2*n)!): seq(denom(a(n)), n=1..20); # Ruperto Corso, Dec 15 2011
a(n) = binomial(2*n+1,n)*(n+1)^2.
(Formerly M4855 N2075)
+0
16
1, 12, 90, 560, 3150, 16632, 84084, 411840, 1969110, 9237800, 42678636, 194699232, 878850700, 3931426800, 17450721000, 76938289920, 337206098790, 1470171918600, 6379820115900, 27569305764000, 118685861314020, 509191949220240, 2177742427450200, 9287309860732800
COMMENTS
Coefficients for numerical differentiation.
Take the first n integers 1,2,3..n and find all combinations with repetitions allowed for the first n of them. Find the sum of each of these combinations to get this sequence. Example for 1 and 2: 1,2,1+1,1+2,2+2 gives sum of 12=a(2). - J. M. Bergot, Mar 08 2016 Let cos(x) = 1 -x^2/2 +x^4/4!-x^6/6!.. = Sum_i (-1)^i x^(2i)/(2i)! be the standard power series of the cosine, and y = 2*(1-cos(x)) = 4*sin^2(x/2) = x^2 -x^4/12 +x^6/360 ...= Sum_i 2*(-1)^(i+1) x^(2i)/(2i)! be a closely related series. Then this sequence represents the reversion x^2 = Sum_i 1/a(i) *y^(i+1). - R. J. Mathar, May 03 2022
REFERENCES
C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 514.
J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 92.
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
R. Shenton and A. W. Kemp, An S-fraction and ln^2(1+x), Journal of Computational and Applied Mathematics, 26 (1989) 367-370 North-Holland.
FORMULA
G.f.: (1 + 2x)/(1 - 4x)^(5/2).
a(n-1) = sum(i_1 + i_2 + ... + i_n) where the sum is over 0 <= i_1 <= i_2 <= ... <= i_n <= n; a(n) = (n+1)^2 C(2n+1, n). - David Callan, Nov 20 2003 Asymptotics: a(n)-> (1/64) * (128*n^2+176*n+41) * 4^n * n^(-1/2)/(sqrt(Pi)), for n->infinity. - Karol A. Penson, Aug 05 2013 E.g.f.: ((1 + 2*x)*(1 + 8*x)*BesselI(0,2*x) + 2*x*(3 + 8*x)*BesselI(1,2*x))*exp(2*x).
Sum_{n>=0} 1/a(n) = Pi^2/9 = A100044. (End) With x = y^2/(1 + y) we have log^2(1 + y) = Sum_{n >= 0} (-1)^n*x^(n+1)/a(n). See Shenton and Kemp.
Series reversion ( Sum_{n >= 0} (-1)^n*x^(n+1)/a(n) ) = Sum_{n >= 1} 2*x^n/(2*n)! = Sum_{n >= 1} x^n/ A002674(n). (End) D-finite with recurrence n^2*a(n) -2*(n+1)*(2*n+1)*a(n-1)=0. - R. J. Mathar, Feb 08 2021
MAPLE
seq((n+1)^2*(binomial(2*n+2, n+1))/2, n=0..29); # Zerinvary Lajos, May 31 2006
MATHEMATICA
Table[Binomial[2n+1, n](n+1)^2, {n, 0, 20}] (* Harvey P. Dale, Mar 23 2011 *)
PROG
(PARI) a(n)=binomial(2*n+1, n)*(n+1)^2
(PARI) x='x+O('x^99); Vec((1+2*x)/(1-4*x)^(5/2)) \\ Altug Alkan, Jul 09 2016 (Python)
from sympy import binomial
def a(n): return binomial(2*n + 1, n)*(n + 1)**2 # Indranil Ghosh, Apr 18 2017
Numerators of coefficients for numerical differentiation.
(Formerly M4034 N1676)
+0
2
1, -5, 259, -3229, 117469, -7156487, 2430898831, -60997921, 141433003757, -25587296781661, 51270597630767, -6791120985104747, 3400039831130408821, -15317460638921852507, 25789165074168004597399, -1550286106708510672406629, 24823277118070193095631689
REFERENCES
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
a(n) is the numerator of (-1)^(n-1)*Cn-1{1^2..(2n-1)^2}/((2n)!*2^(2n-3)), where Cn{1^2..(2n+1)^2} equals 1 when n=0, otherwise it is the sum of the products of all possible combinations, of size n, of the numbers (2k+1)^2 with k=0,1,...,n. - Ruperto Corso, Dec 15 2011
MAPLE
with(combinat):
a:=n->add(mul(k, k=j), j=choose([seq((2*i-1)^2, i=1..n)], n-1))*(-1)^(n-1)/(2^(2*n-3)*(2*n)!):
Denominators of coefficients of log(1+x)/sqrt(1+x).
(Formerly M5139 N2229)
+0
2
1, 1, 24, 12, 640, 1920, 107520, 1792, 2064384, 10321920, 43253760, 64880640, 5398069248, 4198498304, 503819796480, 62977474560, 16610786017280, 4271344975872, 649244436332544, 41618233098240, 27967452642017280
REFERENCES
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).
MATHEMATICA
Denominator[CoefficientList[Series[Log[1 + x]/Sqrt[1 + x]/x, {x, 0, 40}], x]] (* Vincenzo Librandi, Mar 25 2014 *)
Denominators of coefficients in Taylor series expansion of log(1+x)^2/sqrt(1+x).
(Formerly M2133 N0846)
+0
3
1, 1, 2, 24, 48, 5760, 11520, 35840, 215040, 51609600, 103219200, 13624934400, 5449973760, 1322526965760, 3606891724800, 158703235891200, 105802157260800, 14800210341396480, 29600420682792960, 3749386619820441600
COMMENTS
Old title: Denominators of coefficients for numerical differentiation.
REFERENCES
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).
MATHEMATICA
Denominator[CoefficientList[Series[Log[1 + x]^2/Sqrt[1 + x]/x, {x, 0, 40}], x]] (* Vincenzo Librandi, Mar 25 2014 *)
Numerators in expansion of 1/sqrt(1-x).
(Formerly M2508 N0992)
+0
76
1, 1, 3, 5, 35, 63, 231, 429, 6435, 12155, 46189, 88179, 676039, 1300075, 5014575, 9694845, 300540195, 583401555, 2268783825, 4418157975, 34461632205, 67282234305, 263012370465, 514589420475, 8061900920775, 15801325804719
COMMENTS
Also numerator of e(n-1,n-1) (see Maple line).
Leading coefficient of normalized Legendre polynomial.
Common denominator of expansions of powers of x in terms of Legendre polynomials P_n(x).
Also the numerator of binomial(2*n,n)/2^n. - T. D. Noe, Nov 29 2005 This sequence gives the numerators of the Maclaurin series of the Lorentz factor (see Wikipedia link) of 1/sqrt(1-b^2) = dt/dtau where b=u/c is the velocity in terms of the speed of light c, u is the velocity as observed in the reference frame where time t is measured and tau is the proper time. - Stephen Crowley, Apr 03 2007 Truncations of rational expressions like those given by the numerator operator are artifacts in integer formulas and have many disadvantages. A pure integer formula follows. Let n$ denote the swinging factorial and sigma(n) = number of '1's in the base-2 representation of floor(n/2). Then a(n) = (2*n)$ / sigma(2*n) = A056040(2*n) / A060632(2*n+1). Simply said: this sequence is the odd part of the swinging factorial at even indices. - Peter Luschny, Aug 01 2009 The convolution of sequence binomial(2*n,n)/4^n with itself is the constant sequence with all terms = 1.
a(n) equals the denominator of Hypergeometric2F1[1/2, n, 1 + n, -1] (see Mathematica code below). - John M. Campbell, Jul 04 2011 a(n) = numerator of (1/Pi)*Integral_{x=-oo..+oo} 1/(x^2-2*x+2)^n dx. - Leonid Bedratyuk, Nov 17 2012 a(n) = numerator of the mean value of cos(x)^(2*n) from x = 0 to 2*Pi. - Jean-François Alcover, Mar 21 2013 Constant terms for normalized Legendre polynomials. - Tom Copeland, Feb 04 2016 By analytic continuation to the entire complex plane there exist regularized values for divergent sums:
a(n)/ A060818(n) = (-2)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)). Sum_{k>=0} (-1)^k*a(k)/ A060818(k) = 1/sqrt(3). Sum_{k>=0} (-1)^(k+1)*a(k)/ A060818(k) = -1/sqrt(3). a(n)/ A046161(n) = (-1)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)). Sum_{k>=0} (-1)^k*a(k)/ A046161(k) = 1/sqrt(2). Sum_{k>=0} (-1)^(k+1)*a(k)/ A046161(k) = -1/sqrt(2). (End) a(n) = numerator of (1/Pi)*Integral_{x=-oo..+oo} 1/(x^2+1)^n dx. (n=1 is the Cauchy distribution.) - Harry Garst, May 26 2017
Let R(n, d) = (Product_{j prime to d} Pochhammer(j / d, n)) / n!. Then the numerators of R(n, 2) give this sequence and the denominators are A046161. For d = 3 see A273194/ A344402. - Peter Luschny, May 20 2021 Using WolframAlpha, it appears a(n) gives the numerator in the residues of f(z) = 2z choose z at odd negative half integers. E.g., the residues of f(z) at z = -1/2, -3/2, -5/2 are 1/(2*Pi), 1/(16*Pi), and 3/(256*Pi) respectively. - Nicholas Juricic, Mar 31 2022 a(n) is the numerator of (1/Pi) * Integral_{x=-oo..+oo} sech(x)^(2*n+1) dx. The corresponding denominator is A046161. - Mohammed Yaseen, Jul 29 2023 a(n) is the numerator of (1/Pi) * Integral_{x=0..Pi/2} sin(x)^(2*n) dx. The corresponding denominator is A101926(n). - Mohammed Yaseen, Sep 19 2023
REFERENCES
P. J. Davis, Interpolation and Approximation, Dover Publications, 1975, p. 372.
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed. New York: Wiley, 1968; Chap. III, Eq. 4.1.
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).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 6, equation 6:14:6 at page 51.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 102.
FORMULA
a(n) = numerator( binomial(2*n,n)/4^n ) (cf. A046161). a(n) = denominator of (2^n/binomial(2*n,n)). - Artur Jasinski, Nov 26 2011 a(n) = numerator(L(n)), with rational L(n):=binomial(2*n,n)/2^n. L(n) is the leading coefficient of the Legendre polynomial P_n(x).
L(n) = (2*n-1)!!/n! with the double factorials (2*n-1)!! = A001147(n), n >= 0. Numerator in (1-2t)^(-1/2) = 1 + t + (3/2)t^2 + (5/2)t^3 + (35/8)t^4 + (63/8)t^5 + (231/16)t^6 + (429/16)t^7 + ... = 1 + t + 3*t^2/2! + 15*t^3/3! + 105*t^4/4! + 945*t^5/5! + ... = e.g.f. for double factorials A001147 (cf. A094638). - Tom Copeland, Dec 04 2013 a(n)/ A061549(n) = (-1/4)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)). Sum_{k>=0} a(k)/ A061549(k) = 2/sqrt(3). Sum_{k>=0} (-1)^k*a(k)/ A061549(k) = 2/sqrt(5). Sum_{k>=0} (-1)^(k+1)*a(k)/ A061549(k) = -2/sqrt(5). a(n)/ A123854(n) = (-1/2)^n*sqrt(Pi)/(gamma(1/2 - n)*gamma(1 + n)). Sum_{k>=0} a(k)/ A123854(k) = sqrt(2). Sum_{k>=0} (-1)^k*a(k)/ A123854(k) = sqrt(2/3). Sum_{k>=0} (-1)^(k+1)*a(k)/ A123854(k) = -sqrt(2/3). (End)
EXAMPLE
1, 1, 3/2, 5/2, 35/8, 63/8, 231/16, 429/16, 6435/128, 12155/128, 46189/256, ...
binomial(2*n,n)/4^n => 1, 1/2, 3/8, 5/16, 35/128, 63/256, 231/1024, 429/2048, 6435/32768, ...
MAPLE
e := proc(l, m) local k; add(2^(k-2*m)*binomial(2*m-2*k, m-k)*binomial(m+k, m)*binomial(k, l), k=l..m); end;
swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
sigma := n -> 2^(add(i, i=convert(iquo(n, 2), base, 2))):
a := n -> swing(2*n)/sigma(2*n); # (End)
MATHEMATICA
Numerator[ CoefficientList[ Series[1/Sqrt[(1 - x)], {x, 0, 25}], x]]
Table[Denominator[Hypergeometric2F1[1/2, n, 1 + n, -1]], {n, 0, 34}] (* John M. Campbell, Jul 04 2011 *) Numerator[Table[(-2)^n*Sqrt[Pi]/(Gamma[1/2 - n]*Gamma[1 + n]), {n, 0, 20}]] (* Ralf Steiner, Apr 07 2017 *) Numerator[Table[Binomial[2n, n]/2^n, {n, 0, 25}]] (* Vaclav Kotesovec, Apr 07 2017 *) Table[Numerator@LegendreP[2 n, 0]*(-1)^n, {n, 0, 25}] (* Andres Cicuttin, Jan 22 2018 *) A = {1}; Do[A = Append[A, 2^IntegerExponent[n, 2]*(2*n - 1)*A[[n]]/n], {n, 1, 25}]; Print[A] (* John Lawrence, Jul 17 2020 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( pollegendre(n), n) * 2^valuation((n\2*2)!, 2))};
(PARI) a(n)=binomial(2*n, n)>>hammingweight(n); \\ Gleb Koloskov, Sep 26 2021 @CachedFunction
def swing(n):
if n == 0: return 1
return swing(n-1)*n if is_odd(n) else 4*swing(n-1)/n
(Magma)
A001790:= func< n | Numerator((n+1)*Catalan(n)/4^n) >;
CROSSREFS
Cf. A060818 (denominator of binomial(2*n,n)/2^n), A061549 (denominators). Cf. A161198 (triangle of coefficients for (1-x)^((-1-2*n)/2)). Cf. A163590 (odd part of the swinging factorial). First column and diagonal 1 of triangle A100258.
KEYWORD
nonn,easy,nice,frac,changed
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