A255931 a(n) is the numerator of Gamma(n+1/2)^2/(2*n*Pi), the value of an integral with sinh in the denominator.

1, 9, 75, 11025, 178605, 36018675, 2608781175, 4108830350625, 131939107925625, 85734032330071125, 17185776480709711875, 33334677780416604466875, 4807886218329317951953125, 6509191098729563747237109375

Offset

1,2

Links

Table of n, a(n) for n=1..14.

MathOverflow, An identity involving an infinite integral with a sinh in the denominator

Formula

Integral_{-infinity..infinity} (prod_{j=1..n-1} j^2+x^2)*x/sinh(2*Pi*x) dx = Gamma(n+1/2)^2/(2*n*Pi).

The n-th fraction also equals the n-th coefficient in the expansion of 2F1(1/2,1/2; 1; x) * n!*(n-1)!/2.

Example

1/8, 9/64, 75/128, 11025/2048, 178605/2048, 36018675/16384, 2608781175/32768, ...

Mathematica

a[n_] := Gamma[n+1/2]^2/(2*n*Pi) // Numerator; Array[a, 15]
Table[(2*n)!^2 / (n * 2^(4*n+1) * n!^2), {n, 1, 20}] // Numerator (* Vaclav Kotesovec, Mar 11 2015 *)

Crossrefs

Cf. A255932 (denominators).

Sequence in context: A161736 A226180 A274657 * A319957 A056339 A056329

Adjacent sequences: A255928 A255929 A255930 * A255932 A255933 A255934

Keyword

frac,nonn

Author

Jean-François Alcover, Mar 11 2015

Status

approved