A120920
G.f. satisfies: A(x) = G(x)^3 * A(x^4*G(x)^9), where G(x) is the g.f. of the number of ternary trees (A001764): G(x) = 1 + x*G(x)^3.
Original entry on oeis.org
1, 3, 12, 55, 276, 1464, 8058, 45543, 262626, 1538607, 9130446, 54761628, 331403447, 2021021082, 12407102937, 76611488305, 475493441604, 2964664310319, 18560063203353, 116621922800283, 735236268006654
Offset: 0
A(x) = 1 + 3*x + 12*x^2 + 55*x^3 + 276*x^4 + 1464*x^5 + 8058*x^6 +...
= G(x)^3 * A(x^4*G(x)^9) where
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
is g.f. of A001764: G(x) = 1 + x*G(x)^3.
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{a(n)=local(A=1+x,G=(1/x*serreverse(x/(1+3*x+3*x^2+x^3+x*O(x^n))))^(1/3)); for(i=0,n,A=G^3*subst(A,x,x^4*G^9 +x*O(x^n)));polcoeff(A,n,x)}
A120922
Main diagonal of triangle A120919 (cascadence of (1+x)^3); a(n) = A120919(n,n) for n>=0.
Original entry on oeis.org
1, 3, 18, 128, 876, 6138, 43373, 307857, 2194731, 15698743, 112614054, 809905638, 5838361138, 42178611879, 305340946455, 2214760026120, 16094665727934, 117171115942752, 854506665035841, 6242259681316251, 45674776431331398
Offset: 0
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{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); polcoeff(polcoeff(A,n,x),n,y)}
A120923
Row sums of triangle A120919 (cascadence of (1+x)^3).
Original entry on oeis.org
1, 10, 89, 755, 6261, 51276, 416802, 3371901, 27192291, 218814309, 1758106311, 14110481670, 113160495179, 906973579067, 7266174714391, 58193602100496, 465947698757267, 3730070760926851, 29856161486307842, 238947353750059666
Offset: 0
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{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))}
Showing 1-3 of 3 results.
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