A370150 Expansion of g.f. (1 - theta_4(x))/2 / Product_{n>=1} (1 - x^(2*n))^3.

1, 0, 3, -1, 9, -3, 22, -9, 52, -22, 111, -51, 230, -108, 451, -222, 861, -432, 1587, -819, 2861, -1501, 5028, -2691, 8679, -4707, 14691, -8089, 24492, -13638, 40202, -22653, 65141, -37060, 104199, -59863, 164833, -95484, 257920, -150646, 399681, -235141, 613557, -363543, 933869

Offset

1,3

Comments

Column 0 of triangle A370040. The g.f. of triangle A370040, F(x,y), satisfies Sum_{n=-oo..+oo} (-1)^n * (x^n + y*F(x,y))^n = 1 + (y+2)*Sum_{n>=1} (-1)^n * x^(n^2). The g.f. of this sequence is F(x,y) at y = 0.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..202

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Formula

a(n) = A370040(n,0) for n >= 1.

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) A(x) = (1 - theta_4(x))/2 / Product_{n>=1} (1 - x^(2*n))^3.

(2) A(x) = P(x)/Q(x) where P(x) = Sum_{n>=1} (-1)^(n-1) * x^(n^2) and Q(x) = Sum_{n>=0} (-1)^n * (2*n+1) * x^(n*(n+1)).

(3) A(x) = F(x,0) where F(x,y) is the g.f. of triangle A370040 (see comment).

Example

G.f.: A(x) = x + 3*x^3 - x^4 + 9*x^5 - 3*x^6 + 22*x^7 - 9*x^8 + 52*x^9 - 22*x^10 + 111*x^11 - 51*x^12 + 230*x^13 - 108*x^14 + 451*x^15 - 222*x^16 + ...
which equals A(x) = P(x) / Q(x)
where
P(x) = x - x^4 + x^9 - x^16 + x^25 - x^36 + x^49 + ...
Q(x) = 1 - 3*x^2 + 5*x^6 - 7*x^12 + 9*x^20 - 11*x^30 + 13*x^42 + ...

Prog

(PARI) {a(n) = my(P = sum(m=1, sqrtint(n+1), (-1)^(m-1) * x^(m^2) +x*O(x^n)),
Q = sum(m=0, sqrtint(n+1), (-1)^m * (2*m+1) * x^(m*(m+1)) +x*O(x^n)));
polcoeff(P/Q, n)}
for(n=1, 50, print1(a(n), ", "))

Crossrefs

Cf. A370040, A370151, A370152.

Cf. A370153 (dual).

Sequence in context: A328475 A126186 A368568 * A162852 A054448 A106516

Adjacent sequences: A370147 A370148 A370149 * A370151 A370152 A370153

Keyword

sign

Author

Paul D. Hanna, Feb 10 2024

Status

approved