A258783 Numerators of a rational sequence related to the closed form evaluation of a Gaussian integer zeta sum.

1, 1, 2, 1, 2, 2, 4, 223, 854, 4762, 1444, 2324506, 35548, 140343676, 21047728264, 88824427, 160465442, 96633020222386, 18457536052, 1397584483920886, 885721299987868, 2758893844640044, 3793843972393624, 56271391915038457502, 480348904858674456484, 85734822774933179463764, 2140418040120050844508958552

Offset

1,3

Links

Table of n, a(n) for n=1..27.

Jonathan Borwein, Also Eisenstein

Formula

p(1) = 1/5, p(n) = (3/((4n+1)*(2n-3)))*Sum_{k=1..n-1} p(k)*p(n-k).

The closed form of the Gaussian integer zeta sum in question is zeta_G(4n) = p(n)*L^(4n)/(4n-1), where L is the lemniscate constant.

Example

Sequence of fractions begins:
1/5, 1/75, 2/4875, 1/82875, 2/6215625, 2/242409375, 4/19527421875, 223/ 44815433203125, ...

Mathematica

p[1] = 1/5; p[n_] := p[n] = (3/((4n+1)*(2n-3)))*Sum[p[k]*p[n-k], {k, 1, n-1}]; Table[p[n] // Numerator, {n, 1, 30}]

Crossrefs

Cf. A062539 (lemniscate constant), A258784 (denominators).

Sequence in context: A089408 A350287 A208888 * A079318 A050315 A128978

Adjacent sequences: A258780 A258781 A258782 * A258784 A258785 A258786

Keyword

nonn,frac,easy

Author

Jean-François Alcover, Jun 10 2015

Status

approved