A143437 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x))^5.

1, 1, 5, 40, 405, 4745, 61551, 862050, 12831835, 200874055, 3282575310, 55693595381, 977058059380, 17668078651755, 328497282637520, 6267311264123850, 122498870023756800, 2449635783413544555, 50061311067746399725, 1044531750427750075150, 22233430278290842445120

Offset: 0

Author

Paul D. Hanna, Aug 14 2008

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 40*x^3 + 405*x^4 + 4745*x^5 + 61551*x^6 +...
A(x*A(x)) = 1 + x + 6*x^2 + 55*x^3 + 620*x^4 + 7940*x^5 + 111166*x^6 +...
A(x*A(x))^5 = 1 + 5*x + 40*x^2 + 405*x^3 + 4745*x^4 + 61551*x^5 +...
		

Crossrefs

Cf. A143426, A143435, A143436.

Programs

PARI

{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A^5,x,x*A));polcoeff(A,n)}

PARI

a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(n-j+k, j)/(n-j+k)*a(n-j, 5*j))); \\ Seiichi Manyama, Jun 05 2025

Formula

G.f. satisfies: x - G(x) = G(x)^2*A(x)^5 where G(x*A(x)) = x.

From Seiichi Manyama, Jun 05 2025: (Start)

Let a(n,k) = [x^n] A(x)^k.

a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(n-j+k,j)/(n-j+k) * a(n-j,5*j). (End)