A352302 - OEIS

%I #21 Aug 21 2024 11:24:12

%S 1,-1,3,-13,73,-521,4441,-44185,502545,-6429169,91393201,-1429101521,

%T 24378097129,-450504733849,8965682806809,-191174795868841,

%U 4348171177591201,-105077942935229537,2688685949077138657,-72618903735812907553,2064598911185525708601

%N Expansion of e.g.f. 1/(exp(x) - x^2).

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

%F a(n) ~ n! * (-1)^n / ((1 + LambertW(1/2)) * 2^(n+3) * LambertW(1/2)^(n+2)). - _Vaclav Kotesovec_, Mar 12 2022

%F a(n) = n! * Sum_{k=0..floor(n/2)} (-k-1)^(n-2*k)/(n-2*k)!. - _Seiichi Manyama_, Aug 21 2024

%t m = 20; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - x^2), {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^2)))

%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, 2);

%Y Cf. A089148, A352303, A352304.

%Y Cf. A346269.

%K sign

%O 0,3

%A _Seiichi Manyama_, Mar 11 2022