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 A380842 #14 May 29 2025 04:13:05
%S A380842 1,1,10,141,2776,70045,2157156,78452521,3290644288,156380715801,
%T A380842 8304267312100,487328231729581,31318669850761008,2187567259278425557,
%U A380842 165011952533314548676,13368463736048341225425,1157693100510102752463616,106719312722496774534400177,10433609651067618426072766020
%N A380842 Main diagonal of the array A380841.
%F A380842 a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(k+n-1,k)/(n-k)!. - _Seiichi Manyama_, Feb 06 2025
%F A380842 a(n) ~ r^(n + 1/2) * (1+r)^n * n^n / (sqrt(1 + 2*r - r^2) * exp(n) * (1-r)^n), where r = 0.760359234033398901446642379997259705906638343193092252797... is the root of the equation exp(1-r)*(1-r^2)^r = r^(2*r). - _Vaclav Kotesovec_, May 29 2025
%t A380842 A380841[n_,k_]:=n!SeriesCoefficient[1/(1-x*Exp[x])^k,{x,0,n}]; Table[A380841[n,n],{n,0,18}]
%o A380842 (PARI) a(n) = n!*sum(k=0, n, k^(n-k)*binomial(k+n-1, k)/(n-k)!); \\ _Seiichi Manyama_, Feb 06 2025
%Y A380842 Cf. A213643, A380841.
%K A380842 nonn
%O A380842 0,3
%A A380842 _Stefano Spezia_, Feb 05 2025