A326089 G.f. A(x) satisfies: 1 + 2 * Sum_{n>=1} A(x)^(n*(n+1)/2) * x^n = Sum_{n>=0} (1 + x*A(x)^n)^n * x^n.

1, 2, 10, 90, 1002, 12422, 164866, 2294842, 33092066, 490458214, 7430814938, 114644436410, 1796058433818, 28510908961974, 457838834794898, 7427992562185162, 121633693137277970, 2008722124583739830, 33435242160622759594, 560667079009101397162, 9468303108907658924874, 160991659768738107957670, 2755748012982878460654370, 47485639148930596873348890

Offset: 0

Author

Paul D. Hanna, Jun 28 2019

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 90*x^3 + 1002*x^4 + 12422*x^5 + 164866*x^6 + 2294842*x^7 + 33092066*x^8 + 490458214*x^9 + 7430814938*x^10 + ...
such that the following sum
B(x) = 1 + 2*A(x)*x + 2*A(x)^3*x^2 + 2*A(x)^6*x^3 + 2*A(x)^10*x^4 + 2*A(x)^15*x^5 + 2*A(x)^21*x^6 + 2*A(x)^28*x^7 + ... + 2*A(x)^(n*(n+1)/2)*x^n + ...
equals
B(x) = 1 + (1 + x*A(x))*x + (1 + x*A(x)^2)^2*x^2 + (1 + x*A(x)^3)^3*x^3 + (1 + x*A(x)^4)^4*x^4 + (1 + x*A(x)^5)^5*x^5 + ... + (1 + x*A(x)^n)^n*x^n + ...
as well as
B(x) = 1/(1 - x) + A(x)*x/(1 - x*A(x))^2 + A(x)^4*x^2/(1 - x*A(x)^2)^3 + A(x)^9*x^3/(1 - x*A(x)^3)^4 + A(x)^16*x^4/(1 - x*A(x)^4)^5 + ...
where
B(x) = 1 + 2*x + 6*x^2 + 34*x^3 + 290*x^4 + 3082*x^5 + 37078*x^6 + 482122*x^7 + 6611538*x^8 + 94256914*x^9 + 1384318518*x^10 + ...
		

Programs

PARI

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

Formula

The g.f. A(x) allows for the following sums to be equal:

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

(2) B(x) = Sum_{n>=0} (1 + x*A(x)^n)^n * x^n.

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