A352602 a(n) = 4^n*(2^(2*n+1)-1)*(2*n)!.

1, 56, 11904, 5852160, 5274501120, 7606429286400, 16070664624537600, 46802060374022553600, 179724025424120905728000, 879933863508054097526784000, 5350005543376937290448240640000, 39547255119844566012586402775040000, 349281388446657765223160470894018560000

Offset: 0

Author

Artur Jasinski, Mar 22 2022

Keywords

• nonn

Comments

For n>0, PolyGamma(2*n,1/4) = -a(n)*Zeta(2*n+1) - A000816(n)*Pi^(2n+1) = -2^(2*n-1)*(A331839(n)*Zeta(2*n+1) + A000364(n)*Pi^(2n+1)).

Examples

			PolyGamma(2,1/4) = -56*zeta(3) - 2*Pi^3
PolyGamma(4,1/4) = -11904*zeta(5) - 40*Pi^5
PolyGamma(6,1/4) = -5852160*zeta(7) - 1952*Pi^7
		

Crossrefs

Cf. A000302, A000364, A000816, A010050, A331839.

Programs

Maple

A352602 := proc(n)
    4^n*(2^(2*n+1)-1)*(2*n)! ;
end proc:
seq(A352602(n),n=0..30) ; # R. J. Mathar, Aug 19 2022

Mathematica

Table[4^n*(2^(2*n + 1) - 1)*(2*n)!, {n, 0, 12}]

PARI

a(n) = n<<=1; my(f=n!<Kevin Ryde, Mar 23 2022

Formula

a(n) = (-Pi^(2*n+1)*A000816(n) - PolyGamma(2*n,1/4))/zeta(2*n+1).

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

D-finite with recurrence a(n) -40*n*(2*n-1)*a(n-1) +256*n*(n-1)*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 19 2022