A002209 Denominators of coefficients for numerical integration.
1, 2, 12, 8, 720, 288, 60480, 17280, 3628800, 89600, 95800320, 17418240, 2615348736000, 402361344000, 4483454976000, 98402304, 32011868528640000, 342372925440000, 51090942171709440000, 5377993912811520000, 33720021833328230400000
Offset: 0
Examples
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.
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]
Programs
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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 *) -
Maxima
a(n):=denom(((-1)^(n)*sum(stirling1(n+1,k+1)/(k+1),k,0,n))/(n)!); /* Vladimir Kruchinin, Oct 12 2016 */
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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
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
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