A255931 - OEIS

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

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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