A380842 Main diagonal of the array A380841.
1, 1, 10, 141, 2776, 70045, 2157156, 78452521, 3290644288, 156380715801, 8304267312100, 487328231729581, 31318669850761008, 2187567259278425557, 165011952533314548676, 13368463736048341225425, 1157693100510102752463616, 106719312722496774534400177, 10433609651067618426072766020
Offset: 0
Keywords
Programs
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Mathematica
A380841[n_,k_]:=n!SeriesCoefficient[1/(1-x*Exp[x])^k,{x,0,n}]; Table[A380841[n,n],{n,0,18}]
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PARI
a(n) = n!*sum(k=0, n, k^(n-k)*binomial(k+n-1, k)/(n-k)!); \\ Seiichi Manyama, Feb 06 2025
Formula
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(k+n-1,k)/(n-k)!. - Seiichi Manyama, Feb 06 2025
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