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A263188 Coefficient of y^0 in G(x,y)^3 where G(x,y) = Sum_{n=-oo..+oo} (1-x^n)^n * x^n * y^n. 1
1, -6, 3, 6, 12, -12, 3, -30, 18, -48, 63, -30, 51, -36, 129, -168, 147, -78, -42, -120, 279, -372, 195, -180, 339, -294, 351, -588, 969, -444, 90, -840, 1617, -888, 477, -2082, 2487, -1164, 669, -2136, 2526, -1926, 1512, -3402, 7233, -5790, 765, -5154, 12258, -1698, -5754, -8496, 14553, -3444, 9333, -23886, 18234, -6330, 4677, -11544, 15765, -9456, 7059, -23730, 42054, -45486, 23634, -22452, 59061, -16626, -46389, -48780, 104931, -6048, 45039, -136458, 69543, -62280, 98535, -29742, 48996, -93156 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Compare to the coefficient of y^0 in G(x,y)^2, which equals theta_4(x) = 1 - 2*x + 2*x^4 - 2*x^9 + 2*x^16 - 2*x^25 +...+ 2*(-x)^(n^2) +..., where G(x,y) = Sum_{n=-oo..+oo} (1-x^n)^n * x^n * y^n.

Note the curious identity: Sum_{n=-oo..+oo} (1-x^n)^n * x^n = 0.

LINKS

Table of n, a(n) for n=0..81.

EXAMPLE

G.f.: A(x) = 1 - 6*x + 3*x^2 + 6*x^3 + 12*x^4 - 12*x^5 + 3*x^6 - 30*x^7 + 18*x^8 - 48*x^9 + 63*x^10 - 30*x^11 + 51*x^12 - 36*x^13 + 129*x^14 - 168*x^15 +...

such that A(x) is the coefficient of y^0 in G(x,y)^3 where

G(x,y) = N(x,y) + P(x,y), with

P(x,y) = 1 + x*y*(1-x) + (x*y)^2*(1-x^2)^2 + (x*y)^3*(1-x^3)^3 + (x*y)^4*(1-x^4)^4 + (x*y)^5*(1-x^5)^5 + (x*y)^6*(1-x^6)^6 +...+ (x*y)^n*(1-x^n)^n +...

N(x,y) = (-1/y)/(1-x) + (x/y)^2/(1-x^2)^2 + (-x^2/y)^3/(1-x^3)^3 + (x^3/y)^4/(1-x^4)^4 + (-x^4/y)^5/(1-x^5)^5 +...+ (-x^(n-1)/y)^n/(1-x^n)^n +...

PROG

(PARI) {a(n) = my(A=sum(m=-sqrtint(n)-1, n+1, x^m*(1-x^m)^m*y^m +x*O(x^n))); polcoeff(polcoeff(A^3, 0, y), n, x)}

for(n=0, 80, print1(a(n), ", "))

CROSSREFS

Cf. A263189.

Sequence in context: A293560 A021161 A137275 * A268618 A024563 A319232

Adjacent sequences:  A263185 A263186 A263187 * A263189 A263190 A263191

KEYWORD

sign

AUTHOR

Paul D. Hanna, Nov 05 2015

STATUS

approved

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Last modified June 23 18:45 EDT 2021. Contains 345402 sequences. (Running on oeis4.)