%I #17 May 12 2021 03:54:20

%S 0,1,3,10,34,104,200,-400,-2800,85120,2163840,30240000,285331200,

%T 1247769600,-15759744000,-392840448000,-1587505920000,123384188928000,

%U 4345827053568000,82159404687360000,749393890172928000,-10215518583029760000,-570363673124044800000,-9916595355495628800000

%N Expansion of e.g.f. log(1 + Sum_{k>=1} (k+1)*(k+2)/6 * x^k).

%F a(n) = (1/6) * ((n+2)! - Sum_{k=1..n-1} binomial(n-1,k) * (k+2)! * a(n-k)).

%F From _Vaclav Kotesovec_, May 12 2021: (Start)

%F E.g.f: log(2/3 + 1/(3*(1 - x)^3)).

%F 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)

%t 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 *)

%t nmax = 25; CoefficientList[Series[Log[2/3 + 1/(3*(1 - x)^3)], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, May 12 2021 *)

%o (PARI) {a(n) = if (n<1, 0, ((n+2)!-sum(k=1, n-1, binomial(n-1, k)*(k+2)!*a(n-k)))/6)}

%Y Column k=3 of A308497.

%K sign

%O 0,3

%A _Seiichi Manyama_, Jun 01 2019