A369699 a(n) = ( Product_{k=2..n-1} k^k )^2*2^n.

1, 2, 4, 128, 186624, 24461180928, 477757440000000000, 2079947912569159680000000000, 2821337329257037336798707056640000000000, 1588271718091057966534018222437593205367111680000000000, 476782128558831092865939388401540382200008655926882902537666560000000000

Offset

0,2

Comments

PolyGamma(-2, n) can be expressed as log(a(n)*Pi^n)/2 - (n^2 - n)/2.

Links

Table of n, a(n) for n=0..10.

Index entries for sequences related to factorial numbers

Formula

a(n) = Product_{k=0..n-1} 2*k^(2*k), with 2*k^(2*k) = 2 for k = 0.

a(n) = A002109(n-1)^2*2^n for n > 0.

a(n) = ((n-1)!)^(2*(n-1))*2^n/G(n)^2, for n > 0 where G() is the Barnes G-function.

a(n) = exp(2*(Integral_{x=0..n} log(Gamma(x)) dx) + n*(n-1))/Pi^n.

a(n) = exp(2*PolyGamma(-2, n) + n*(n-1))/Pi^n, where PolyGamma(-2, n) is the -2nd derivative of the digamma function.

zeta'(-1, n) - zeta'(-1) = (log(a(n)) - n*log(2))/2 = Sum_{m=1..n-1} m*log(m), where zeta'(-1, n) is the derivative of the Hurwitz zeta function and zeta'(-1) the derivative of the Riemann zeta function evaluated at -1.

Mathematica

a[n_] := ((n-1)!)^(2*(n-1))*2^n / BarnesG[n]^2; a[0] = 1; Array[a, 11, 0] (* Amiram Eldar, Jan 29 2024 *)

Prog

(PARI) a(n) = prod(k=0, n-1, 2*k^(2*k))
(PARI) a(n) = prod(k=2, n-1, k^k)^2*2^n

Crossrefs

Cf. A002109, A059332, A172286, A290770.

Sequence in context: A134710 A009073 A326213 * A012561 A012558 A012617

Adjacent sequences: A369696 A369697 A369698 * A369700 A369701 A369702

Keyword

nonn

Author

Thomas Scheuerle, Jan 29 2024

Status

approved