A205800 Expansion of e.g.f. exp( Sum_{n>=1} x^(n^2) ).

1, 1, 1, 1, 25, 121, 361, 841, 21841, 547345, 4541041, 23292721, 169658281, 7550279881, 95230199065, 692107448761, 25431412450081, 563675083228321, 9791797014753121, 112525775579561185, 3370231071632996281, 65798618669268652441, 1345746844683430533961

Offset: 0

Author

Paul D. Hanna, Jan 31 2012

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + x^3/3! + 25*x^4/4! + 121*x^5/5! +...
where
log(A(x)) = x + x^4 + x^9 + x^16 + x^25 + x^36 + x^49 + x^64 +...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..448

Crossrefs

Cf. A193375, A205801, A205802, A329256.

Programs

Maple

seq(coeff(series(factorial(n)*(exp(add(x^(k^2),k=1..n))),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Oct 23 2018

Mathematica

With[{nn=30},CoefficientList[Series[Exp[Sum[x^n^2,{n,nn}]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 01 2020 *)

PARI

{a(n)=n!*polcoeff(exp(sum(m=1, sqrtint(n+1), x^(m^2)+x*O(x^n))), n)}

PARI

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

Formula

E.g.f.: exp((theta_3(x) - 1)/2), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Oct 23 2018

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..floor(sqrt(n))} k^2 * a(n-k^2)/(n-k^2)!. - Seiichi Manyama, Apr 29 2022