A341378 G.f. A(x) satisfies: A(x) = P(x)/Q(x) where P(x) = Sum_{n>=0} (n+1)*x^n*(1 + x*A(x)^n)^n and Q(x) = Sum_{n>=0} x^n*(1 + x*A(x)^n)^n.

1, 1, 2, 4, 13, 49, 198, 847, 3796, 17661, 84754, 417556, 2104652, 10825545, 56714218, 302185765, 1635727560, 8987357912, 50090647729, 283058794945, 1621221821820, 9409181197452, 55327487031279, 329591827614844, 1989063796779401

Offset: 0

Author

Paul D. Hanna, Feb 19 2021

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 13*x^4 + 49*x^5 + 198*x^6 + 847*x^7 + 3796*x^8 + 17661*x^9 + 84754*x^10 + 417556*x^11 + 2104652*x^12 + ...
such that A(x) = P(x)/Q(x) where
P(x) = 1 + 2*x*(1 + x*A(x)) + 3*x^2*(1 + x*A(x)^2)^2 + 4*x^3*(1 + x*A(x)^3)^3 + 5*x^4*(1 + x*A(x)^4)^4 + 6*x^5*(1 + x*A(x)^5)^5 + ...
Q(x) = 1 + x*(1 + x*A(x)) + x^2*(1 + x*A(x)^2)^2 + x^3*(1 + x*A(x)^3)^3 + x^4*(1 + x*A(x)^4)^4 + x^5*(1 + x*A(x)^5)^5 + ...
also
Q(x) = 1/(1 - x) + x^2*A(x)/(1 - x*A(x))^2 + x^4*A(x)^4/(1 - x*A(x)^2)^3 + x^6*A(x)^9/(1 - x*A(x)^3)^4 + x^8*A(x)^16/(1 - x*A(x)^4)^5 + ...
explicitly,
P(x) = 1 + 2*x + 5*x^2 + 12*x^3 + 36*x^4 + 124*x^5 + 471*x^6 + 1918*x^7 + 8246*x^8 + 37024*x^9 + 172266*x^10 + 825870*x^11 + 4062450*x^12 + ...
Q(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 35*x^5 + 125*x^6 + 484*x^7 + 1994*x^8 + 8625*x^9 + 38804*x^10 + 180360*x^11 + 861833*x^12 + ...
		

Crossrefs

Cf. A337851.

Programs

PARI

{a(n) = my(A=1 +x*O(x^n)); for(i=1,n,
A = sum(m=0,n, (m+1) * x^m *(1 + x*A^m +x*O(x^n))^m ) / sum(m=0,n, x^m *(1 + x*A^m +x*O(x^n))^m )   );
polcoeff(H=A,n)}
for(n=0,20,print1(a(n),", "))

Formula

G.f. A(x) satisfies:

(1) A(x) = P(x)/Q(x) where

P(x) = Sum_{n>=0} (n+1) * x^n * (1 + x*A(x)^n)^n and

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

(2) A(x) = P(x)/Q(x) where

P(x) = Sum_{n>=0} (n+1) * x^n * (1 + x*A(x)^n)^n and

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