A353223 Expansion of e.g.f. (1 - x^3)^(-1/x^2).

1, 1, 1, 1, 13, 61, 181, 2101, 19321, 107353, 1338121, 18021961, 153519301, 2162889301, 37434929533, 437750929981, 7054260835441, 146656527486001, 2197288472426641, 40414798347009553, 970905798377330941, 17791752518018762221, 370864149434372540101

Offset

0,5

Links

Table of n, a(n) for n=0..22.

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..floor((n+2)/3)} (3*k-2)/k * a(n-3*k+2)/(n-3*k+2)!.

a(n) = n! * Sum_{k=0..floor(n/3)} |Stirling1(n-2*k,n-3*k)|/(n-2*k)!.

a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (3*exp(n)). - Vaclav Kotesovec, May 04 2022

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^3)^(-1/x^2)))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-log(1-x^3)/x^2)))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, (i+2)\3, (3*j-2)/j*v[i-3*j+3]/(i-3*j+2)!)); v;
(PARI) a(n) = n!*sum(k=0, n\3, abs(stirling(n-2*k, n-3*k, 1))/(n-2*k)!);

Crossrefs

Cf. A121452, A246689, A353222, A353225.

Sequence in context: A361657 A308461 A357964 * A302560 A252970 A047673

Adjacent sequences: A353220 A353221 A353222 * A353224 A353225 A353226

Keyword

nonn

Author

Seiichi Manyama, May 01 2022

Status

approved