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

%I

%S 0,1,0,0,-1,0,0,4,0,-1,-22,0,13,140,0,-136,-970,9,1330,7104,-231,

%T -12650,-54096,3900,118780,423890,-54810,-1108380,-3393696,695640,

%U 10311840,27615648,-8282604,-95810606,-227480848,94449456,889817328,1890685212,-1044402840,-8263944216,-15811484852

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

%C 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.

%H Max Alekseyev, <a href="/A259938/b259938.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SeriesReversion.html">Series Reversion</a>

%F 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

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

%o (PARI) Vec( serreverse( sum(i=1,32,x^i^2) + O(x^33^2) ) ); \\ _Max Alekseyev_, Jul 06 2021

%Y Cf. A049140, A192540.

%K easy,sign

%O 0,8

%A _Vladimir Reshetnikov_, Jul 09 2015

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Last modified September 21 07:14 EDT 2021. Contains 347596 sequences. (Running on oeis4.)