A239109 Number of hybrid 6-ary trees with n internal nodes.

1, 2, 23, 375, 7138, 148348, 3262975, 74673216, 1759690865, 42412172598, 1040644972314, 25907046248766, 652763779424538, 16614703783094140, 426563932954831827, 11033640140115676862, 287265076610919864178, 7522060666571155198520, 197969862318742854908470

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..300

SeoungJi Hong and SeungKyung Park, Hybrid d-ary trees and their generalization, Bull. Korean Math. Soc. 51 (2014), No. 1, pp. 229-235. See p. 233.

Formula

From Paul D. Hanna, Mar 30 2014: (Start)

G.f. A(x) satisfies:

(1) A(x) = (1 + x*A(x)^5) * (1 + x*A(x)^6).

(2) A(x) = ( (1/x)*Series_Reversion( x*(1-x-x^2)^5/(1+x)^5 ) )^(1/5).

(3) A(x) = exp( Sum_{n>=1} x^n*A(x)^(4*n)/n * Sum_{k=0..n} C(n,k)^2 * A(x)^k ).

(4) A(x) = exp( Sum_{n>=1} x^n*A(x)^(5*n)/n * Sum_{k=0..n} C(n,k)^2 / A(x)^k ).

(5) A(x) = Sum_{n>=0} Fibonacci(n+2) * x^n * A(x)^(5*n).

(6) A(x) = G(x*A(x)^4) where G(x) = A(x/G(x)^4) is the g.f. of A007863 (number of hybrid binary trees with n internal nodes).

The formal inverse of g.f. A(x) is (sqrt(1-2*x+5*x^2) - (1+x))/(2*x^6).

a(n) = [x^n] ( (1+x)/(1-x-x^2) )^(5*n+1) / (5*n+1).

(End)

Mathematica

(1/x InverseSeries[x*(1 - x - x^2)^5/(1 + x)^5 + O[x]^20])^(1/5) // CoefficientList[#, x]& (* Jean-François Alcover, Oct 02 2019 *)

Prog

(PARI) a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=(1 + x*A^5)*(1 + x*A^6)); polcoeff(A, n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 30 2014
(PARI) a(n)=polcoeff( ((1/x)*serreverse( x*(1-x-x^2)^5/(1+x +x*O(x^n))^5))^(1/5), n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 30 2014
(PARI) a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*A^j)*x^m*A^(4*m)/m))); polcoeff(A, n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 30 2014
(PARI) a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2/A^j)*x^m*A^(5*m)/m))); polcoeff(A, n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 30 2014
(PARI) a(n)=polcoeff(((1+x)/(1-x-x^2 +x*O(x^n)))^(5*n+1)/(5*n+1), n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 30 2014

Crossrefs

Cf. A000045, A007863, A215654, A239107, A239108, A239109.

Column k=6 of A245049.

Sequence in context: A277830 A197740 A234868 * A266923 A060941 A338178

Adjacent sequences: A239106 A239107 A239108 * A239110 A239111 A239112

Keyword

nonn

Author

N. J. A. Sloane, Mar 26 2014

Status

approved