Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #14 Sep 08 2015 13:55:19
%S 1,4294967296
%N Product_{k=1..n} k^(k^5).
%C The next terms: a(3) has 126 digits, a(4) has 743 digits, a(5) has 2927 digits.
%C In general, product_{k=1..n} k^(k^m) ~ A(m) * n^(B(m+1)/(m+1) + sum_{j=1..n} j^m) * exp(-n^(m+1)/(m+1)^2 + sum_{j=1..m-1} (1/(j+1) * B(j+1) * binomial(m,j) * n^(m-j) * sum_{i=0..j-1} 1/(m-i) )), where A(m) is the generalized Glaisher-Kinkelin constant (see A074962, A243262, A243263, A243264, A243265), and B(n) is the Bernoulli number A027641(n) / A027642(n).
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.
%H Vaclav Kotesovec, <a href="/A255344/b255344.txt">Table of n, a(n) for n = 1..4</a>
%F a(n) ~ A243265 * n^(n^2*(n+1)^2*(2*n^2+2*n-1)/12 + 1/252) / exp(47*n^2/720 - n^4/12 + n^6/36).
%t Table[Product[k^(k^5), {k, 1, n}], {n, 1, 5}]
%o (PARI) a(n)=prod(k=2,n,k^k^5) \\ _Charles R Greathouse IV_, Sep 08 2015
%Y Cf. A002109, A051675, A255321, A255323, A243265.
%K nonn
%O 1,2
%A _Vaclav Kotesovec_, Feb 21 2015