A359669 a(n) = coefficient of x^n in A(x) where x = Sum_{n=-oo..+oo} (-1)^(n-1) * x^(n*(n+1)) * A(x)^(n^2).

1, 1, 0, 3, 6, 13, 55, 142, 429, 1495, 4538, 14894, 50279, 164189, 554402, 1883870, 6371434, 21854442, 75183191, 259137380, 899092908, 3127293679, 10907931688, 38188033950, 133998312862, 471339759941, 1662075700667, 5872497411731, 20790187564837, 73741279736768

Offset: 0

Author

Paul D. Hanna, Jan 17 2023

Keywords

• nonn

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A359672.

Programs

PARI

{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(x - sum(m=-#A, #A, (-1)^(m-1) * x^(m*(m+1)) * Ser(A)^(m^2) ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

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

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