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

%I A205800 #22 Apr 30 2022 08:21:06
%S A205800 1,1,1,1,25,121,361,841,21841,547345,4541041,23292721,169658281,
%T A205800 7550279881,95230199065,692107448761,25431412450081,563675083228321,
%U A205800 9791797014753121,112525775579561185,3370231071632996281,65798618669268652441,1345746844683430533961
%N A205800 Expansion of e.g.f. exp( Sum_{n>=1} x^(n^2) ).
%H A205800 Seiichi Manyama, <a href="/A205800/b205800.txt">Table of n, a(n) for n = 0..448</a>
%F A205800 E.g.f.: exp((theta_3(x) - 1)/2), where theta_3() is the Jacobi theta function. - _Ilya Gutkovskiy_, Oct 23 2018
%F A205800 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
%e A205800 E.g.f.: A(x) = 1 + x + x^2/2! + x^3/3! + 25*x^4/4! + 121*x^5/5! +...
%e A205800 where
%e A205800 log(A(x)) = x + x^4 + x^9 + x^16 + x^25 + x^36 + x^49 + x^64 +...
%p A205800 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
%t A205800 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 *)
%o A205800 (PARI) {a(n)=n!*polcoeff(exp(sum(m=1, sqrtint(n+1), x^(m^2)+x*O(x^n))), n)}
%o A205800 (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
%Y A205800 Cf. A193375, A205801, A205802, A329256.
%K A205800 nonn
%O A205800 0,5
%A A205800 _Paul D. Hanna_, Jan 31 2012