A370153 Expansion of g.f. (theta_3(x) - 1)/2 * Product_{n>=1} (1 - x^(4*n-2)) / (1 - x^(4*n)).

1, 0, -1, 1, 1, -1, -2, 1, 4, -2, -5, 3, 6, -4, -9, 6, 13, -8, -17, 11, 21, -15, -28, 19, 39, -25, -49, 33, 60, -42, -78, 53, 101, -68, -125, 87, 153, -108, -192, 134, 241, -167, -295, 207, 357, -255, -438, 311, 540, -380, -652, 465, 781, -563, -946, 678, 1145, -819, -1368, 986, 1627

Offset: 1

Author

Paul D. Hanna, Feb 10 2024

Keywords

• sign

Comments

Column 0 of triangle A370041. The g.f. of triangle A370041, G(x,y), satisfies Sum_{n=-oo..+oo} (x^n - y*G(x,y))^n = 1 - (y-2)*Sum_{n>=1} x^(n^2). The g.f. of this sequence is G(x,y) at y = 0.

Examples

			G.f.: A(x) = x - x^3 + x^4 + x^5 - x^6 - 2*x^7 + x^8 + 4*x^9 - 2*x^10 - 5*x^11 + 3*x^12 + 6*x^13 - 4*x^14 - 9*x^15 + 6*x^16 + 13*x^17 - 8*x^18 + ...
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 + x^2 + x^6 + x^12 + x^20 + x^30 + x^42 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..201
• Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Crossrefs

Cf. A370041, A370154, A370155.

Cf. A370150 (dual).

Programs

PARI

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

Formula

a(n) = A370041(n,0) for n >= 1.

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

(1) A(x) = (theta_3(x) - 1)/2 * Product_{n>=1} (1 - x^(4*n-2))/(1 - x^(4*n)).

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

(3) A(x) = G(x,0) where G(x,y) is the g.f. of triangle A370041 (see comment).