A355364 G.f. A(x) satisfies: x^2*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

1, 0, 1, 3, 11, 34, 110, 350, 1147, 3800, 12836, 43929, 152285, 533205, 1883187, 6698612, 23974179, 86258459, 311811314, 1131863444, 4124127216, 15078422405, 55301519095, 203405409935, 750122683729, 2773048061073, 10274442343829, 38147288401915

Offset: 0

Author

Paul D. Hanna, Jul 19 2022

Keywords

• nonn

Comments

Equals the antidiagonal sums of A355360; a(n) = Sum_{k=0..n} A355360(n-k,k).

Examples

			G.f.: A(x) = 1 + x^2 + 3*x^3 + 11*x^4 + 34*x^5 + 110*x^6 + 350*x^7 + 1147*x^8 + 3800*x^9 + 12836*x^10 + 43929*x^11 + 152285*x^12 + ...
where
x^2*A(x) = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x)^n + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A355360, A355361, A355362, A355363, A355365.

Programs

PARI

{a(n) = my(A=[1,0],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*n+9));
A[#A] = polcoeff( x^2*Ser(A) - sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:

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

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

(3) x^2*A(x)*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.

a(n) ~ c * d^n / n^(3/2), where d = 3.92217771004386918... and c = 0.52890084997249... - Vaclav Kotesovec, Jul 03 2025