A355362 G.f. A(x) satisfies: 2*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

1, 2, 14, 106, 852, 7286, 65216, 603714, 5731930, 55506348, 546091942, 5443033448, 54845812094, 557774491672, 5717718435034, 59017814463718, 612873311614338, 6398538141213916, 67121038262747380, 707114126290890810, 7478082640450505012, 79360375914717108922

Offset: 0

Author

Paul D. Hanna, Jul 19 2022

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 106*x^3 + 852*x^4 + 7286*x^5 + 65216*x^6 + 603714*x^7 + 5731930*x^8 + 55506348*x^9 + 546091942*x^10 + ...
where
2*x*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, A355363, A355364, A355365.

Programs

PARI

{a(n) = my(A=[1,2],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*n+9));
A[#A] = polcoeff( 2*x*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

a(n) = Sum_{k=0..n} A355360(n,k) * 2^k for n >= 0.

G.f. A(x) satisfies:

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

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

(3) 2*x*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 = 11.38813717738101179115221618346020026348459... and c = 0.5257715220992591718905720654742321646... - Vaclav Kotesovec, Jul 03 2025