This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A369699 #35 Apr 20 2024 09:50:53
%S A369699 1,2,4,128,186624,24461180928,477757440000000000,
%T A369699 2079947912569159680000000000,
%U A369699 2821337329257037336798707056640000000000,1588271718091057966534018222437593205367111680000000000,476782128558831092865939388401540382200008655926882902537666560000000000
%N A369699 a(n) = ( Product_{k=2..n-1} k^k )^2*2^n.
%C A369699 PolyGamma(-2, n) can be expressed as log(a(n)*Pi^n)/2 - (n^2 - n)/2.
%H A369699 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A369699 a(n) = Product_{k=0..n-1} 2*k^(2*k), with 2*k^(2*k) = 2 for k = 0.
%F A369699 a(n) = A002109(n-1)^2*2^n for n > 0.
%F A369699 a(n) = ((n-1)!)^(2*(n-1))*2^n/G(n)^2, for n > 0 where G() is the Barnes G-function.
%F A369699 a(n) = exp(2*(Integral_{x=0..n} log(Gamma(x)) dx) + n*(n-1))/Pi^n.
%F A369699 a(n) = exp(2*PolyGamma(-2, n) + n*(n-1))/Pi^n, where PolyGamma(-2, n) is the -2nd derivative of the digamma function.
%F A369699 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.
%t A369699 a[n_] := ((n-1)!)^(2*(n-1))*2^n / BarnesG[n]^2; a[0] = 1; Array[a, 11, 0] (* _Amiram Eldar_, Jan 29 2024 *)
%o A369699 (PARI) a(n) = prod(k=0, n-1, 2*k^(2*k))
%o A369699 (PARI) a(n) = prod(k=2, n-1, k^k)^2*2^n
%Y A369699 Cf. A002109, A059332, A172286, A290770.
%K A369699 nonn
%O A369699 0,2
%A A369699 _Thomas Scheuerle_, Jan 29 2024