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%I #23 Aug 19 2022 05:01:05
%S 1,56,11904,5852160,5274501120,7606429286400,16070664624537600,
%T 46802060374022553600,179724025424120905728000,
%U 879933863508054097526784000,5350005543376937290448240640000,39547255119844566012586402775040000,349281388446657765223160470894018560000
%N a(n) = 4^n*(2^(2*n+1)-1)*(2*n)!.
%C 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)).
%F a(n) = (-Pi^(2*n+1)*A000816(n) - PolyGamma(2*n,1/4))/zeta(2*n+1).
%F a(n) = 2^(2*n-1)*A331839(n).
%F 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
%e PolyGamma(2,1/4) = -56*zeta(3) - 2*Pi^3
%e PolyGamma(4,1/4) = -11904*zeta(5) - 40*Pi^5
%e PolyGamma(6,1/4) = -5852160*zeta(7) - 1952*Pi^7
%p A352602 := proc(n)
%p 4^n*(2^(2*n+1)-1)*(2*n)! ;
%p end proc:
%p seq(A352602(n),n=0..30) ; # _R. J. Mathar_, Aug 19 2022
%t Table[4^n*(2^(2*n + 1) - 1)*(2*n)!, {n, 0, 12}]
%o (PARI) a(n) = n<<=1; my(f=n!<<n); f<<(n+1) - f; \\ _Kevin Ryde_, Mar 23 2022
%Y Cf. A000302, A000364, A000816, A010050, A331839.
%K nonn
%O 0,2
%A _Artur Jasinski_, Mar 22 2022
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