A258784 Denominators of a rational sequence related to the closed form evaluation of a Gaussian integer zeta sum.

5, 75, 4875, 82875, 6215625, 242409375, 19527421875, 44815433203125, 7185407790234375, 1699625304228515625, 22095128954970703125, 1538152402200285498046875, 1024661605286766357421875, 177310518163637910787353515625, 1171727007531373860453094482421875

Offset

1,1

Links

Table of n, a(n) for n=1..15.

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] // Denominator, {n, 1, 30}]

Crossrefs

Cf. A062539 (lemniscate constant), A258783 (numerators).

Sequence in context: A238560 A303125 A332714 * A051481 A277296 A364323

Adjacent sequences: A258781 A258782 A258783 * A258785 A258786 A258787

Keyword

nonn,frac,easy

Author

Jean-François Alcover, Jun 10 2015

Status

approved