%I #16 Aug 21 2024 11:23:58

%S 1,-1,1,5,-47,239,-239,-11761,170689,-1237825,-2360159,238756319,

%T -4146035519,32586126143,359988680689,-18567245926321,351652342984321,

%U -2283764958280321,-89760640709677247,3866819337993369023,-74731210747948586879,167887841949213912959

%N Expansion of e.g.f. 1/(exp(x) - x^3).

%F a(n) = n * (n-1) * (n-2) * a(n-3) - Sum_{k=1..n} binomial(n,k) * a(n-k) for n > 2.

%F a(n) = n! * Sum_{k=0..floor(n/3)} (-k-1)^(n-3*k)/(n-3*k)!. - _Seiichi Manyama_, Aug 21 2024

%t m = 21; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - x^3), {x, 0, m}], x] (* _Amiram Eldar_, Mar 12 2022 *)

%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)-x^3)))

%o (PARI) b(n, m) = if(n==0, 1, sum(k=1, n, (-1+(k==m)*m!)*binomial(n, k)*b(n-k, m)));

%o a(n) = b(n, 3);

%Y Cf. A089148, A352302, A352304.

%Y Cf. A352299.

%K sign

%O 0,4

%A _Seiichi Manyama_, Mar 11 2022