OFFSET
0,3
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
a(n) = A191578(2*n,n)/(2*n)!, n > 0. - Vladimir Kruchinin, Feb 02 2013
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).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..100
S. Akiyama and Y. Tanigawa, Multiple zeta values at non-positive integers, Ramanujan J. 5 (2001), 327-351.
D. Kruchinin and V. Kruchinin, A Generating Function for the Diagonal T2n,n in Triangles, Journal of Integer Sequence, Vol. 18 (2015), article 15.4.6.
Guodong Liu, Some computational formulas for Norlund numbers, Fib. Quart., 45 (2007), 133-137.
A. N. Lowan and H. Salzer, Table of coefficients in numerical integration formulas, J. Math. Phys., 22 (1943), 49-50.
A. N. Lowan and H. Salzer, Table of coefficients in numerical integration formulas, J. Math. Phys. Mass. Inst. Tech. 22 (1943), 49-50.[Annotated scanned copy]
M. O. Rubinstein, Identities for the Riemann zeta function, Ramanujan J. 27, No. 1, 29-42 (2012) and arXiv:0812.2592.
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
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
CROSSREFS
KEYWORD
frac,nonn,easy,nice
AUTHOR
STATUS
approved