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%I A002209 M2015 N0796 #69 Jan 05 2025 19:51:32
%S A002209 1,2,12,8,720,288,60480,17280,3628800,89600,95800320,17418240,
%T A002209 2615348736000,402361344000,4483454976000,98402304,32011868528640000,
%U A002209 342372925440000,51090942171709440000,5377993912811520000,33720021833328230400000
%N A002209 Denominators of coefficients for numerical integration.
%C A002209 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
%C A002209 The numerators are given in A002208.
%D A002209 Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 529.
%D A002209 N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924.
%D A002209 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002209 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002209 T. D. Noe, <a href="/A002209/b002209.txt">Table of n, a(n) for n = 0..100</a>
%H A002209 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 A002209 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 A002209 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 A002209 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]
%F A002209 G.f. of A002208(n)/a(n): -x/((1-x)*log(1-x)).
%F A002209 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
%F A002209 a(n) = denominator(((-1)^n/n!)*Sum_{k=0..n} Stirling1(n+1,k+1)/(k+1)). - _Vladimir Kruchinin_, Oct 12 2016
%e A002209 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.
%t A002209 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 *)
%t A002209 Denominator[CoefficientList[Series[-x/((1-x)Log[1-x]),{x,0,20}],x]] (* _Harvey P. Dale_, Feb 01 2013 *)
%o A002209 (Maxima)
%o A002209 a(n):=denom(((-1)^(n)*sum(stirling1(n+1,k+1)/(k+1),k,0,n))/(n)!); /* _Vladimir Kruchinin_, Oct 12 2016 */
%o A002209 (Python)
%o A002209 from math import factorial
%o A002209 from fractions import Fraction
%o A002209 from sympy.functions.combinatorial.numbers import stirling
%o A002209 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
%Y A002209 Cf. A002208. See also A002657, A002790, A002206, A002207, A006232, A006233.
%K A002209 nonn,frac,easy,nice
%O A002209 0,2
%A A002209 _N. J. A. Sloane_

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