A002209 Denominators of coefficients for numerical integration. (Formerly M2015 N0796)

1, 2, 12, 8, 720, 288, 60480, 17280, 3628800, 89600, 95800320, 17418240, 2615348736000, 402361344000, 4483454976000, 98402304, 32011868528640000, 342372925440000, 51090942171709440000, 5377993912811520000, 33720021833328230400000

Offset

0,2

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).

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.

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]

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

Crossrefs

Cf. A002208. See also A002657, A002790, A002206, A002207, A006232, A006233.

Sequence in context: A001898 A268230 A229628 * A100654 A166544 A081468

Adjacent sequences: A002206 A002207 A002208 * A002210 A002211 A002212

Keyword

nonn,frac,easy,nice

Author

N. J. A. Sloane

Status

approved