A002208 Numerators of coefficients for numerical integration.
1, 1, 5, 3, 251, 95, 19087, 5257, 1070017, 25713, 26842253, 4777223, 703604254357, 106364763817, 1166309819657, 25221445, 8092989203533249, 85455477715379, 12600467236042756559, 1311546499957236437, 8136836498467582599787
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
- 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.
- Index entries for sequences related to Bernoulli numbers.
Programs
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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
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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 *) -
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 */
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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
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
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