A320898 Expansion of e.g.f. exp(theta_3(x) - 1), where theta_3() is the Jacobi theta function.

1, 2, 4, 8, 64, 512, 2944, 13568, 134656, 2371328, 29676544, 268141568, 2560761856, 53154824192, 991944441856, 13085180592128, 187309143556096, 4400237083492352, 105779411022905344, 1939709049732595712, 37680665654471950336, 882429584512554893312, 23052947736212625424384

Offset: 0

Author

Ilya Gutkovskiy, Oct 23 2018

Keywords

• nonn

Crossrefs

Cf. A000122, A205800.

Programs

Maple

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

Mathematica

nmax = 22; CoefficientList[Series[Exp[EllipticTheta[3, 0, x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[SquaresR[1, k] k! Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]

Formula

E.g.f.: exp(2*Sum_{k>=1} x^(k^2)).

a(0) = 1; a(n) = Sum_{k=1..n} A000122(k)*k!*binomial(n-1,k-1)*a(n-k).