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A369699
a(n) = ( Product_{k=2..n-1} k^k )^2*2^n.
0
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.
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
KEYWORD
nonn
AUTHOR
Thomas Scheuerle, Jan 29 2024
STATUS
approved