A143500 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^2).

1, 1, 1, 3, 10, 46, 244, 1481, 10020, 74400, 599573, 5200284, 48223360, 475557054, 4965035754, 54672110310, 632853655686, 7678552433184, 97404631390960, 1288861146261679, 17752479062092470, 254051633672160696, 3770953046565933003, 57964955567444706668

Offset: 0

Author

Paul D. Hanna, Aug 20 2008

Keywords

• nonn

Examples

			G.f. A(x) = 1 + x + x^2 + 3*x^3 + 10*x^4 + 46*x^5 + 244*x^6 +...
A(x)^2 = 1 + 2*x + 3*x^2 + 8*x^3 + 27*x^4 + 118*x^5 + 609*x^6 +...
A(x*A(x)^2) = 1 + x + 3*x^2 + 10*x^3 + 46*x^4 + 244*x^5 +...
If G(x*A(x)^2) = x then
G(x) = x - 2*x^2 + 5*x^3 - 18*x^4 + 68*x^5 - 300*x^6 + 1283*x^7 -+...
A(G(x)) = 1 + A(x)*G(x) = (x/G(x))^(1/2) where
A(x)*G(x) = x - x^2 + 4*x^3 - 12*x^4 + 59*x^5 - 209*x^6 + 1199*x^7 -...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..230

Crossrefs

Cf. A087949, A143501.

Programs

PARI

{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,x*A^2));polcoeff(A,n)}

PARI

a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-2*j+k, j)/(2*n-2*j+k)*a(n-j, j))); \\ Seiichi Manyama, Jun 04 2025

Formula

G.f. satisfies: G(x) = x/[1 + A(x)*G(x)]^2 = x/A(G(x))^2 where G(x*A(x)^2) = x.

From Seiichi Manyama, Jun 04 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(2*n-2*j+k,j)/(2*n-2*j+k) * a(n-j,j). (End)