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 A361281 #21 Mar 13 2023 11:34:44
%S A361281 1,1,5,37,481,10001,288901,10820965,511186817,29843419681,
%T A361281 2106779832901,176180844038981,17165338119936865,1924030148121500017,
%U A361281 245630480526435293381,35409038825312233143301,5719025066628373334423041,1027649751647068260334391105
%N A361281 a(n) = n! * Sum_{k=0..n} binomial(n*k,n-k)/k!.
%C A361281 From _Peter Bala_, Mar 12 2023: (Start)
%C A361281 It appears that a(n) == 1 (mod 4) and a(5*n+2) == 0 (mod 5) for all n. More generally we conjecture that a(n+k) == a(n) (mod k) for all n and k. If true, then for each k, the sequence a(n) taken modulo k is a periodic sequence and the period divides k.
%C A361281 Let F(x) and G(x) be power series with integer coefficients with G(0) = 1. Define b(n) = n! * [x^n] F(x)*exp(x*G(x)^n). Then we conjecture that b(n+k) == b(n) (mod k) for all n and k. The present sequence is the case F(x) = 1, G(x) = 1 + x. Cf. A278070. (End)
%H A361281 Seiichi Manyama, <a href="/A361281/b361281.txt">Table of n, a(n) for n = 0..275</a>
%F A361281 a(n) = n! * [x^n] exp(x * (1+x)^n).
%F A361281 log(a(n)) ~ n*(2*log(n) - log(log(n)) - 1 - log(2) + log(log(n))/log(n) + 1/(2*log(n)) + log(2)/log(n) - 1/(8*log(n)^2)). - _Vaclav Kotesovec_, Mar 12 2023
%o A361281 (PARI) a(n) = n!*sum(k=0, n, binomial(n*k, n-k)/k!);
%Y A361281 Main diagonal of A361277.
%Y A361281 Cf. A096131, A099237, A226391, A278070, A293013.
%K A361281 nonn
%O A361281 0,3
%A A361281 _Seiichi Manyama_, Mar 06 2023