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A259938 Expansion of the series reversion of Sum_{n>=1} x^(n^2). 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,8

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

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

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A049140, A192540.

Sequence in context: A195739 A323128 A334703 * A342202 A136452 A247703

Adjacent sequences:  A259935 A259936 A259937 * A259939 A259940 A259941

KEYWORD

easy,sign

AUTHOR

Vladimir Reshetnikov, Jul 09 2015

STATUS

approved

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Last modified August 1 00:13 EDT 2021. Contains 346377 sequences. (Running on oeis4.)