A380641 Expansion of e.g.f. exp(x/(1 - 3*x)^3).
1, 1, 19, 379, 8857, 244801, 7904251, 292980619, 12257946289, 570627408097, 29212843607011, 1629314013114811, 98250285167099209, 6365331315043185889, 440712959779710869707, 32460639303987670526731, 2533396174719346231613281, 208776665140069914314618689
Offset: 0
Programs
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Mathematica
Table[n! * Sum[3^k * Binomial[3*n-2*k-1,k]/(n-k)!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Jan 29 2025 *) -
PARI
a(n) = n!*sum(k=0, n, 3^k*binomial(3*n-2*k-1, k)/(n-k)!);
Formula
a(n) = n! * Sum_{k=0..n} 3^k * binomial(3*n-2*k-1,k)/(n-k)!.
E.g.f.: exp( Sum_{k>=1} k*(k+1)/2 * 3^(k-1) * x^k ).
From Vaclav Kotesovec, Jan 29 2025: (Start)
a(n) = (12*n-11)*a(n-1) - 6*(n-1)*(9*n-19)*a(n-2) + 108*(n-3)*(n-2)*(n-1)*a(n-3) - 81*(n-4)*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 3^n * n^(n - 1/8) / (2 * exp(n - 4*n^(3/4)/3 - sqrt(n)/6 + n^(1/4)/72 + 1/81)) * (1 + 16957/(207360*n^(1/4))). (End)