A302121 Denominators of a series converging to Euler's constant.

4, 96, 72, 46080, 1152, 2322432, 100352, 7431782400, 2090188800, 2452488192000, 2697737011200, 64274810535936000, 2923954176000, 1799694695006208000, 3085190905724928, 33566877054287216640000, 4458100858772520960000, 120538655501945394954240000, 1057781497894797312000

Offset

1,1

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..250

Ia. V. Blagouchine, Three Notes on Ser's and Hasse's Representations for the Zeta-functions. INTEGERS, Electronic Journal of Combinatorial Number Theory, vol. 18A, Article #A3, pp. 1-45, 2018. arXiv:1606.02044 [math.NT], 2016.

Formula

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.

Example

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

Maple

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

Mathematica

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}]

Prog

(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!))

Crossrefs

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

Sequence in context: A210857 A220682 A210926 * A188895 A068114 A272109

Adjacent sequences: A302118 A302119 A302120 * A302122 A302123 A302124

Keyword

frac,nonn

Author

Iaroslav V. Blagouchine, Apr 01 2018

Status

approved