A363557 Expansion of g.f. A(x) satisfying 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * A(x)^n / Product_{k=1..n+1} (1 - x^k).

1, 1, 1, 2, 4, 9, 20, 47, 112, 273, 677, 1702, 4330, 11128, 28847, 75341, 198066, 523713, 1391869, 3716098, 9962252, 26806275, 72372721, 195994320, 532266707, 1449216287, 3955193019, 10818202369, 29650108510, 81417795070, 223964216673, 617097850848, 1702943168118

Offset: 0

Author

Paul D. Hanna, Jul 11 2023

Keywords

• nonn

Comments

Related identities:

(1) 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * B(x)^n / Product_{k=1..n+1} (1 - x^k*B(x)), where B(x) = 1/(1-x).

(2) 1 = Sum_{n>=0} (-1)^n * x^(n*(n+1)/2) / Product_{k=1..n+1} (1 - x^k).

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 20*x^6 + 47*x^7 + 112*x^8 + 273*x^9 + 677*x^10 + 1702*x^11 + 4330*x^12 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A082395, A363555.

Programs

PARI

{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( sum(m=0,2*sqrtint(#A), (-1)^m * (x)^(m*(m-1)/2) * Ser(A)^m / prod(k=1,m+1, (1 - x^k +x*O(x^#A) ) )),#A-1);); A[n+1]}
for(n=0,32,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.

(1) 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * A(x)^n / Product_{k=1..n+1} (1 - x^k).

(2) 1/(A(x) - x) = Sum_{n>=0} (-1)^n * x^(n*(n+1)/2) * A(x)^n / Product_{k=1..n+1} (1 - x^k).