A354651 G.f. A(x) satisfies: 1/(1 - x) = Sum_{n>=1} (-1)^(n-1) * A(x)^(n^2).

1, 1, 1, 2, 5, 11, 25, 64, 168, 434, 1136, 3046, 8246, 22400, 61290, 169036, 468628, 1304390, 3646104, 10232796, 28814306, 81376616, 230462906, 654363034, 1862260359, 5311064061, 15176758091, 43448083792, 124593820615, 357853635931, 1029326055479, 2964817204082

Offset: 1

Author

Paul D. Hanna, Jun 18 2022

Keywords

• nonn

Comments

Conjectures:

(C.1) a(4*n) = 0 (mod 2) for n >= 0.

(C.2) a(4*n+1) = a(4*n+2) = a(4*n+3) (mod 2) for n >= 0.

(C.3) a(4*n+1) = a(4*n+3) (mod 4) for n >= 0.

Examples

			G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 5*x^5 + 11*x^6 + 25*x^7 + 64*x^8 + 168*x^9 + 434*x^10 + 1136*x^11 + 3046*x^12 + 8246*x^13 + 22400*x^14 + ...
where
1/(1-x) = A(x) - A(x)^4 + A(x)^9 - A(x)^16 + A(x)^25 - A(x)^36 + A(x)^49 -+ ... + (-1)^(n-1) * A(x)^(n^2) + ...
By the Jacobi triple product
(1 - 3*x)/(1 - x) = (1 - A(x)^2)*(1 - A(x))^2 * (1 - A(x)^4) * (1 - A(x)^3)^2 * (1 - A(x)^6) * (1 - A(x)^5)^2 * (1 - A(x)^8) * (1 - A(x)^7)^2 * ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..625
• Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Crossrefs

Cf. A006456 (the series reversion of -A(-x) is the g.f. for A006456, apart from the initial term).

Cf. A355151.

Programs

PARI

{a(n) = my(A=[0,1],t); for(i=1,n, A = concat(A,0); t = sqrtint(#A)+1;
A[#A] = 1 + polcoeff( sum(n=1,t, (-1)^n * Ser(A)^(n^2)), #A-1)); H=A; A[n+1]}
for(n=1,40,print1(a(n),", "))

Formula

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

(1) (1 - 3*x)/(1 - x) = Sum_{n=-oo..+oo} (-1)^n * A(x)^(n^2).

(2) (1 - 3*x)/(1 - x) = Product_{n>=1} (1 - A(x)^(2*n)) * (1 - A(x)^(2*n-1))^2, by the Jacobi triple product identity.

(3) (1 - 3*x)^2/(1 - x)^2 = 1 + 4*Sum_{n>=1} (-1)^n * A(x)^(2*n-1)/(1 + A(x)^(2*n-1)), by a q-series identity for the Jacobi theta_3 function.

(4) (1 - 3*x)^4/(1 - x)^4 = 1 + 8*Sum_{n>=1} (-1)^n * n * A(x)^n/(1 + A(x)^n), by a q-series identity for the Jacobi theta_3 function.