A331839 a(n) = (4^(n + 1) - 2)*(2*n)!.

2, 28, 1488, 182880, 41207040, 14856307200, 7847004211200, 5713142135500800, 5484741986820096000, 6713362606110031872000, 10204325758699297505280000, 18857600746080668455403520000, 41637586170036526348967608320000, 108257726461843735266949595136000000, 327371366649945523117538738700288000000

Offset

0,1

Links

Table of n, a(n) for n=0..14.

Eric Weisstein's World of Mathematics, Polygamma Function

Formula

a(n) = -(polygamma(2n, 1/4)/2^(2n-1) + abs(Euler(2n))*Pi^(2n+1))/zeta(2n+1) for n > 0.

a(n) = (2n)!*(2^(2n + 2) - 2).

a(n+1) = (2n)!*Stirling2(2n, 2)/binomial(2n, 2).

a(n) = A010050(n)*(A000302(n+1) - 2). - Omar E. Pol, May 02 2020

Example

polygamma(2, 1/4) = -2^1*(28*zeta(3) + Pi^3),
polygamma(4, 1/4) = -2^3*(1488*zeta(5) + 5*Pi^5),
polygamma(6, 1/4) = -2^5*(182880*zeta(7) + 61*Pi^7),
polygamma(8, 1/4) = -2^7*(41207040*zeta(9) + 1385*Pi^9),
etc.

Mathematica

Table[(2 n + 2)!*StirlingS2[2 n + 2, 2]/Binomial[2 n + 2, 2], {n, 0, 17}]
Prepend[FullSimplify[Table[-(PolyGamma[2 n, 1/4]/2^(2 n - 1) + Abs[EulerE[2 n]] Pi^(2 n + 1))/Zeta[2 n + 1], {n, 1, 16}]], 2]

Crossrefs

Cf. A000364, A000302, A010050.

Sequence in context: A071220 A063794 A361405 * A238817 A202942 A356518

Adjacent sequences: A331836 A331837 A331838 * A331840 A331841 A331842

Keyword

nonn

Author

Artur Jasinski, Jan 29 2020

Extensions

a(0) = 2 and new name by Peter Luschny, May 02 2020.

Status

approved