A302121 - OEIS

%I #20 Oct 29 2018 03:53:47

%S 4,96,72,46080,1152,2322432,100352,7431782400,2090188800,

%T 2452488192000,2697737011200,64274810535936000,2923954176000,

%U 1799694695006208000,3085190905724928,33566877054287216640000,4458100858772520960000,120538655501945394954240000,1057781497894797312000

%N Denominators of a series converging to Euler's constant.

%C gamma = 3/4 - 11/96 - 1/72 - 311/46080 - 5/1152 - 7291/2322432 - ..., see formula (104) in the reference below.

%H G. C. Greubel, <a href="/A302121/b302121.txt">Table of n, a(n) for n = 1..250</a>

%H Ia. V. Blagouchine, <a href="http://math.colgate.edu/~integers/sjs3/sjs3.Abstract.html">Three Notes on Ser's and Hasse's Representations for the Zeta-functions.</a> INTEGERS, Electronic Journal of Combinatorial Number Theory, vol. 18A, Article #A3, pp. 1-45, 2018. <a href="http://arxiv.org/abs/1606.02044">arXiv:1606.02044 [math.NT], 2016</a>.

%F a(n) = Denominators of ((1/2)*(-1)^(n+1)*(Sum_{l=0..n-1} (S_1(n-1,l)*((-1/2)^(l+1) + 1)/(l+1)))/(n!) + (-1)^(n+1)*(Sum_{l=1..n} S_1(n,l)/(l+1)))/(n*n!)), where S_1(x,y) are the signed Stirling numbers of the first kind.

%e Denominators of 3/4, -11/96, -1/72, -311/46080, -5/1152, -7291/2322432, ...

%p a := proc (n) options operator, arrow; denum((1/2)*(-1)^(n+1)*(sum(Stirling1(n-1, l)*((-1/2)^(l+1)+1)/(l+1), l = 0 .. n-1))/factorial(n)+(-1)^(n+1)*(sum(Stirling1(n, l)/(l+1), l = 1 .. n))/(n*factorial(n))) end proc

%t a[n_] := Denominator[(1/2)*(-1)^(n+1)*(Sum[StirlingS1[n-1,l]*((-1/2)^(l+1) + 1)/(l+1),{l,0,n-1}])/(n!) + (-1)^(n+1)*(Sum[StirlingS1[n, l]/(l+1),{l,1,n}])/(n*n!)]; Table[a[n], {n, 1, 24}]

%o (PARI) a(n) = denominator((1/2)*(-1)^(n+1)*(sum(l=0,n-1,stirling(n-1,l)*((-1/2)^(l+1) + 1)/(l+1)))/(n!) + (-1)^(n+1)*(sum(l=1,n,stirling(n,l)/(l+1)))/(n*n!))

%Y Cf. A302120 (numerators of this series), A262856, A262858.

%K frac,nonn

%O 1,1

%A _Iaroslav V. Blagouchine_, Apr 01 2018