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 A293013 #17 Aug 25 2025 06:29:09
%S A293013 1,1,5,55,961,24101,818821,36053515,1984670465,132825475081,
%T A293013 10583425959301,988018789759871,106673677280748865,
%U A293013 13172700275176482925,1842428769970603518341,289406832942160060794451,50677793314733587473331201,9829328870566195730521433105
%N A293013 a(n) = n! * [x^n] exp(x/(1 - x)^n).
%C A293013 Conjecture: 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. - _Peter Bala_, Mar 12 2023
%H A293013 Seiichi Manyama, <a href="/A293013/b293013.txt">Table of n, a(n) for n = 0..274</a>
%F A293013 a(n) = A293012(n,n).
%F A293013 For n > 0, a(n) = n! * Sum_{k=1..n} binomial((n-1)*(k+1), k*n - 1)/k!. - _Vaclav Kotesovec_, Aug 24 2025
%F A293013 log(a(n)) ~ n * (2*log(n) - log(log(n)) - 1 - log(2) + log(log(n))/log(n) + (1 + 2*log(2))/(2*log(n))). - _Vaclav Kotesovec_, Aug 25 2025
%t A293013 Table[n! SeriesCoefficient[Exp[x/(1 - x)^n] , {x, 0, n}], {n, 0, 17}]
%t A293013 (* or *)
%t A293013 nmax = 20; Join[{1}, Table[n!*Sum[Binomial[(n-1)*(k+1), k*n - 1]/k!, {k, 1, n}], {n, 1, nmax}]] (* _Vaclav Kotesovec_, Aug 24 2025 *)
%Y A293013 Main diagonal of A293012. Cf. A361281.
%K A293013 nonn,easy,changed
%O A293013 0,3
%A A293013 _Ilya Gutkovskiy_, Sep 28 2017