A384199 Expansion of e.g.f. log(1 + x)/(1 - 3*x).
0, 1, 5, 47, 558, 8394, 150972, 3171132, 76102128, 2054797776, 61643570400, 2034241452000, 73232652355200, 2856073920854400, 119955098448864000, 5397979517377171200, 259103015526429849600, 13214253812770712217600, 713569705533931031654400
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
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Magma
[0] cat [n le 1 select 1 else 3 * n * Self(n-1) - (-1)^n * Factorial(n-1): n in [1..20]]; // Vincenzo Librandi, May 22 2025
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Mathematica
a[n_]:= n! * Sum[(-1)^(k-1)*3^(n-k)/k,{k,1,n}];Table[a[n],{n,0,18}] (* Vincenzo Librandi, May 22 2025 *) -
PARI
a(n) = n!*sum(k=1, n, (-1)^(k-1)*3^(n-k)/k);
Formula
a(n) = n! * Sum_{k=1..n} (-1)^(k-1) * 3^(n-k)/k.
a(n) = 3 * n * a(n-1) - (-1)^n * (n-1)!.
a(n) = (2*n+1) * a(n-1) + 3 * (n-1)^2 * a(n-2).
a(n) ~ log(4/3) * 3^n * n!. - Vaclav Kotesovec, May 23 2025