A259938 Expansion of the series reversion of Sum_{n>=1} x^(n^2).

0, 1, 0, 0, -1, 0, 0, 4, 0, -1, -22, 0, 13, 140, 0, -136, -970, 9, 1330, 7104, -231, -12650, -54096, 3900, 118780, 423890, -54810, -1108380, -3393696, 695640, 10311840, 27615648, -8282604, -95810606, -227480848, 94449456, 889817328, 1890685212, -1044402840, -8263944216, -15811484852

Offset: 0

Author

Vladimir Reshetnikov, Jul 09 2015

Keywords

• easy
• sign

Comments

x + x^4 + x^9 + x^16 + x^25 + ... is the expansion of (theta_3(0, x) - 1)/2, where theta_3 is the Jacobi theta function.

Links

• Max Alekseyev, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Series Reversion

Crossrefs

Cf. A049140, A192540.

Programs

Mathematica

InverseSeries[(EllipticTheta[3, 0, x] - 1)/2 + O[x]^30][[3]]

PARI

Vec( serreverse( sum(i=1,32,x^i^2) + O(x^33^2) ) ); \\ Max Alekseyev, Jul 06 2021

Formula

For n>1, a(n) = Sum_{j2,j3,...} (-1)^(j2+j3+...) * (n-1+j2+j3+...)! / (j2!*j3!*...) / n!, where the sum is taken over all nonnegative integers j2, j3, ... such that (2^2-1)*j2 + (3^2-1)*j3 + ... = n-1. - Max Alekseyev, Jul 06 2021