A369086 Expansion of g.f. A(x) satisfying 2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * A(x)^n.

1, 2, 2, 12, 22, 144, 314, 2094, 4980, 33962, 85104, 589050, 1526520, 10690958, 28357414, 200513838, 540848844, 3855418084, 10529575602, 75589847732, 208400923746, 1505463570864, 4180703306104, 30372614867534, 84817986727644, 619420989037864, 1737266065061962

Offset: 0

Author

Paul D. Hanna, Jan 22 2024

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 2*x + 2*x^2 + 12*x^3 + 22*x^4 + 144*x^5 + 314*x^6 + 2094*x^7 + 4980*x^8 + 33962*x^9 + 85104*x^10 + 589050*x^11 + 1526520*x^12 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A354645.

Programs

PARI

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

Formula

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

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

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

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