A301335 a(n) = [x^n] 1/(1 + (1/2)*n*(1 - theta_3(x))), where theta_3() is the Jacobi theta function.
1, 1, 4, 27, 260, 3175, 47304, 833147, 16941120, 390611331, 10070060200, 287028156162, 8962583345856, 304255011200647, 11156593415089808, 439452231820920000, 18505340390664634384, 829599437871129843839, 39447684087807950938908, 1983038000428208822539998, 105080571577382659860160800
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Mathematica
Table[SeriesCoefficient[1/(1 + (1/2) n (1 - EllipticTheta[3, 0, x])), {x, 0, n}], {n, 0, 20}] Table[SeriesCoefficient[1/(1 - n Sum[x^k^2, {k, 1, n}]), {x, 0, n}], {n, 0, 20}]
Formula
a(n) = [x^n] 1/(1 - n*Sum_{k>=1} x^(k^2)).
a(n) ~ n^n * (1 + 1/n^2 - 3/n^3 + 1/(2*n^4) - 13/(2*n^5) + 127/(6*n^6) - 4/n^7 + 335/(8*n^8) - 665/(4*n^9) + 337/(15*n^10) + ...). - Vaclav Kotesovec, Mar 19 2018
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