A353180 -id:A353180 - cp's OEIS frontend

A329256 Expansion of e.g.f. exp(Sum_{k>=1} x^(k^2) / (k^2)!).

1, 1, 1, 1, 2, 6, 16, 36, 106, 443, 1796, 6161, 23816, 122266, 643644, 2934296, 14002237, 83835433, 532282819, 3005258539, 17039094646, 115611682810, 848428608644, 5682350940168, 37297365940462, 281594230420802, 2323660209441962, 17929392395804072

Offset: 0

Ilya Gutkovskiy, Nov 09 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..611

Cf. A000110, A010052, A205799, A205800, A205801, A205802, A205804, A353180.

nmax = 27; CoefficientList[Series[Exp[Sum[x^(k^2)/(k^2)!, {k, 1, Floor[nmax^(1/2)] + 1}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Boole[IntegerQ[k^(1/2)]] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 27}]

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, sqrtint(N), x^k^2/(k^2)!)))) \\ Seiichi Manyama, Apr 29 2022

a(n) = if(n==0, 1, sum(k=1, sqrtint(n), binomial(n-1, k^2-1)*a(n-k^2))); \\ Seiichi Manyama, Apr 29 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A010052(k) * a(n-k).

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

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 29 2022

nonn

Cf. A006456, A007840, A353180.

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 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, sqrtint(N), x^k^2/(k^2)))))

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;

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

cp's OEIS Frontend

A329256 Expansion of e.g.f. exp(Sum_{k>=1} x^(k^2) / (k^2)!).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula