A308499 Expansion of e.g.f. log(1 + Sum_{k>=1} (k+1)*(k+2)/6 * x^k).
0, 1, 3, 10, 34, 104, 200, -400, -2800, 85120, 2163840, 30240000, 285331200, 1247769600, -15759744000, -392840448000, -1587505920000, 123384188928000, 4345827053568000, 82159404687360000, 749393890172928000, -10215518583029760000, -570363673124044800000, -9916595355495628800000
Offset: 0
Keywords
Crossrefs
Column k=3 of A308497.
Programs
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Mathematica
a[0] = 0; a[n_] := a[n] = ((n+2)! - Sum[Binomial[n-1,k] * (k+2)! * a[n-k], {k,1,n-1}])/6; Array[a, 24, 0] (* Amiram Eldar, May 12 2021 *) nmax = 25; CoefficientList[Series[Log[2/3 + 1/(3*(1 - x)^3)], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, May 12 2021 *) -
PARI
{a(n) = if (n<1, 0, ((n+2)!-sum(k=1, n-1, binomial(n-1, k)*(k+2)!*a(n-k)))/6)}
Formula
a(n) = (1/6) * ((n+2)! - Sum_{k=1..n-1} binomial(n-1,k) * (k+2)! * a(n-k)).
From Vaclav Kotesovec, May 12 2021: (Start)
E.g.f: log(2/3 + 1/(3*(1 - x)^3)).
a(n) ~ -2^(n/2 + 1) * (n-1)! * cos(n*arctan(sqrt(3)/(1 - 2^(4/3)))) / (2 + 2^(1/3) - 2^(2/3))^(n/2). (End)
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