%I M3737 N1527 #100 Jan 05 2025 19:51:32

%S 1,1,5,3,251,95,19087,5257,1070017,25713,26842253,4777223,

%T 703604254357,106364763817,1166309819657,25221445,8092989203533249,

%U 85455477715379,12600467236042756559,1311546499957236437,8136836498467582599787

%N Numerators of coefficients for numerical integration.

%C The denominators are given in A002209.

%C 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

%C a(n) = A191578(2*n,n)/(2*n)!, n > 0. - _Vladimir Kruchinin_, Feb 02 2013

%C The denominators are given in A002209.

%D E. Isaacson and H. B. Keller, Analysis of Numerical Methods, ISBN 0 471 42865 5, 1966, John Wiley and Sons, pp. 318-319.

%D Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 529.

%D N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A002208/b002208.txt">Table of n, a(n) for n = 0..100</a>

%H S. Akiyama and Y. Tanigawa, <a href="http://math.tsukuba.ac.jp/~akiyama/papers/Mzvnrev.pdf">Multiple zeta values at non-positive integers</a>, Ramanujan J. 5 (2001), 327-351.

%H D. Kruchinin and V. Kruchinin, <a href="http://cs.uwaterloo.ca/journals/JIS/VOL18/Kruchinin/kruch9.html">A Generating Function for the Diagonal T2n,n in Triangles</a>, Journal of Integer Sequence, Vol. 18 (2015), article 15.4.6.

%H Guodong Liu, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/45-2/quartliu02_2007.pdf">Some computational formulas for Norlund numbers</a>, Fib. Quart., 45 (2007), 133-137.

%H A. N. Lowan and H. Salzer, <a href="http://dx.doi.org/10.1002/sapm194322149">Table of coefficients in numerical integration formulas</a>, J. Math. Phys., 22 (1943), 49-50.

%H A. N. Lowan and H. Salzer, <a href="/A002206/a002206.pdf">Table of coefficients in numerical integration formulas</a>, J. Math. Phys. Mass. Inst. Tech. 22 (1943), 49-50.[Annotated scanned copy]

%H M. O. Rubinstein, <a href="https://doi.org/10.1007/s11139-010-9276-8">Identities for the Riemann zeta function</a>, Ramanujan J. 27, No. 1, 29-42 (2012) and <a href="https://arxiv.org/abs/0812.2592">arXiv:0812.2592</a>.

%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers</a>.

%F G.f. of rationals a(n)/A002209(n): -x/((1-x)*log(1-x)).

%F 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]

%F a(n) = numerator((-1)^n*Sum_{k=0..n} (k!*Stirling2(n,k)* Stirling1(n+k,n))/(n+k)!). - _Vladimir Kruchinin_, Feb 02 2013

%F 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

%F 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

%F a(n) = numerator(((-1)^n/n!)*Sum_{k=0..n} Stirling1(n+1,k+1)/(k+1)). - _Vladimir Kruchinin_, Oct 12 2016

%e 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.

%p 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

%t Numerator/@CoefficientList[Series[-x/((1-x)Log[1-x]),{x,0,20}],x] (* _Harvey P. Dale_, May 04 2011 *)

%t 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 *)

%o (Maxima)

%o 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 */

%o a(n):=num(((-1)^(n)*sum(stirling1(n+1,k+1)/(k+1),k,0,n))/(n)!); /* _Vladimir Kruchinin_, Oct 12 2016 */

%o (Python)

%o from math import factorial

%o from fractions import Fraction

%o from sympy.functions.combinatorial.numbers import stirling

%o 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

%Y Cf. A002209. See also A002657, A002790, A006232, A006233, A002206, A002207, A191578.

%K frac,nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_