A120923 Row sums of triangle A120919 (cascadence of (1+x)^3).

1, 10, 89, 755, 6261, 51276, 416802, 3371901, 27192291, 218814309, 1758106311, 14110481670, 113160495179, 906973579067, 7266174714391, 58193602100496, 465947698757267, 3730070760926851, 29856161486307842, 238947353750059666

Offset: 0

Author

Paul D. Hanna, Jul 17 2006

Keywords

• nonn

Crossrefs

Cf. A120919, A120920, A120921, A120922; A001764 (ternary trees).

Programs

PARI

{a(n)=local(A,F=(1+x)^3,d=3,G=x,H=1+x,S=ceil(log(n+1)/log(d+1))); for(i=0,n,G=x*subst(F,x,G+x*O(x^n)));for(i=0,S,H=subst(H,x,x*G^d+x*O(x^n))*G/x); A=(x*H-y*subst(H,x,x*y^d +x*O(x^n)))/(x*subst(F,x,y)-y); sum(k=0,3*n,polcoeff(polcoeff(A,n,x),k,y))}

Formula

G.f.: A(x) = H(x)*(1-x)/(1-8*x), where H(x) is g.f. of A120920: H(x) = G*H(x^4*G^3) and G(x) is g.f. of A001764: G(x) = 1 + x*G(x)^3.