A353184 Expansion of e.g.f. 1/(1 - Sum_{k>=1} x^(k^2) / (k^2)).

1, 1, 2, 6, 30, 180, 1260, 10080, 93240, 1015560, 12146400, 158004000, 2226193200, 34162128000, 565750785600, 10034584560000, 190820565936000, 3845407181616000, 81995523626016000, 1844123531009760000, 43689721287532320000, 1086745683839175360000

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

a(0) = 1; a(n) = Sum_{k=1..floor(sqrt(n))} (k^2-1)! * binomial(n,k^2) * a(n-k^2).

Mathematica

a[0] = 1; a[n_] := a[n] = Sum[(k^2 - 1)! * Binomial[n, k^2] * a[n - k^2], {k, 1, Floor@Sqrt[n]}]; Array[a, 22, 0] (* Amiram Eldar, Apr 30 2022 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, sqrtint(N), x^k^2/(k^2)))))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, sqrtint(i), (j^2-1)!*binomial(i, j^2)*v[i-j^2+1])); v;

Crossrefs

Cf. A006456, A007840, A353180.

Sequence in context: A180895 A180896 A180897 * A096769 A365975 A111059

Adjacent sequences: A353181 A353182 A353183 * A353185 A353186 A353187

Keyword

nonn

Author

Seiichi Manyama, Apr 29 2022

Status

approved