%I #27 Sep 08 2022 08:46:21

%S 3,11,1,311,5,7291,243,14462317,3364621,3337014731,3155743303,

%T 65528247068741,2627553901,1439156737843967,2213381206625,

%U 21757704362231905789,2627003970197650333,64925181492079668050329,523317843775891637,161371847993975070290712761,78461950306245817433389909

%N Absolute value of the numerators 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="/A302120/b302120.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) = abs(Numerators 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 Numerators of 3/4, -11/96, -1/72, -311/46080, -5/1152, -7291/2322432, ...

%p a:= proc(n) abs(numer((1/2)*(-1)^(n+1)*(add(Stirling1(n-1, l)*((-1/2)^(l+1)+1)/(l+1), l = 0 .. n-1))/(n)!+(-1)^(n+1)*(add(Stirling1(n, l)/(l+1), l = 1 .. n))/(n*(n)!))) end proc: seq(a(n), n=1..23);

%t a[n_] := Numerator[(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[Abs[a[n]], {n, 1, 24}]

%o (PARI) a(n) = abs(numerator((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!)))

%o (Magma) [3] cat [Abs(Numerator( (1/2)*(-1)^(n+1)*(&+[StirlingFirst(n-1,k)*((-1/2)^(k+1) + 1)/(k+1): k in [1..n-1]])/Factorial(n) + (-1)^(n+1)*(&+[StirlingFirst(n,k)/(k+1): k in [1..n]])/(n*Factorial(n)) )): n in [2..30]]; // _G. C. Greubel_, Oct 29 2018

%Y Cf. A302121 (denominators of this series), A262856, A262858.

%K frac,nonn

%O 1,1

%A _Iaroslav V. Blagouchine_, Apr 01 2018