A143501 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^3).

1, 1, 1, 4, 16, 92, 616, 4729, 40776, 388057, 4028230, 45207583, 544680014, 7004865885, 95694153485, 1382946630490, 21067128029388, 337224872043659, 5656357906530796, 99168643108816180, 1813250965008114981, 34506927801196386939, 682210688659107549765, 13988252589131863518730

Offset: 0

Author

Paul D. Hanna, Aug 20 2008

Keywords

• nonn

Examples

			G.f. A(x) = 1 + x + x^2 + 4*x^3 + 16*x^4 + 92*x^5 + 616*x^6 + 4729*x^7 +...
A(x)^3 = 1 + 3*x + 6*x^2 + 19*x^3 + 78*x^4 + 411*x^5 + 2617*x^6 +...
A(x*A(x)^3) = 1 + x + 4*x^2 + 16*x^3 + 92*x^4 + 616*x^5 + 4729*x^6 +...
If G(x*A(x)^3) = x then
G(x) = x - 3*x^2 + 12*x^3 - 64*x^4 + 372*x^5 - 2385*x^6 + 15675*x^7 -+...
A(G(x)) = 1 + A(x)*G(x) = (x/G(x))^(1/3) where
A(x)*G(x) = x - 2*x^2 + 10*x^3 - 51*x^4 + 324*x^5 - 1985*x^6 + 13938*x^7 -...
		

Crossrefs

Cf. A087949, A143500.

Programs

PARI

{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,x*A^3));polcoeff(A,n)}

PARI

a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-3*j+k, j)/(3*n-3*j+k)*a(n-j, j))); \\ Seiichi Manyama, Jun 04 2025

Formula

G.f. satisfies: G(x) = x/[1 + A(x)*G(x)]^3 = x/A(G(x))^3 where G(x*A(x)^3) = 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(3*n-3*j+k,j)/(3*n-3*j+k) * a(n-j,j). (End)