A239107 Number of hybrid 4-ary trees with n internal nodes.

1, 2, 15, 155, 1854, 24124, 331575, 4736345, 69616637, 1046054129, 15995716263, 248111418112, 3894303176880, 61737213540306, 987116931080661, 15899835212249761, 257758369219909534, 4202381519278498915, 68859442092723799788, 1133401910867109123200

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..800

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.

Sheng-liang Yang and Mei-yang Jiang, Pattern avoiding problems on the hybrid d-trees, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. (in Mandarin)

Formula

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

G.f. A(x) satisfies:

(1) A(x) = (1 + x*A(x)^3) * (1 + x*A(x)^4).

(2) A(x) = ( (1/x)*Series_Reversion( x*(1-x-x^2)^3/(1+x)^3 ) )^(1/3).

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

(4) A(x) = exp( Sum_{n>=1} x^n*A(x)^(3*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)^(3*n).

(6) A(x) = G(x*A(x)^2) where G(x) = A(x/G(x)^2) 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^4).

a(n) = [x^n] ( (1+x)/(1-x-x^2) )^(3*n+1) / (3*n+1).

(End)

a(n) = 1/(3*n+1) * Sum_{i=0..n} C(3*n+i,i)*C(3*n+i+1,n-i). - Alois P. Heinz, Jul 10 2014

Mathematica

(1/x InverseSeries[x(1 - x - x^2)^3/(1 + x)^3 + O[x]^21])^(1/3) // 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^3)*(1 + x*A^4)); 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)^3/(1+x +x*O(x^n))^3))^(1/3), 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^(2*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^(3*m)/m))); polcoeff(A, n) \\ Paul D. Hanna, Mar 30 2014
for(n=0, 20, print1(a(n), ", "))
(PARI) a(n)=polcoeff(((1+x)/(1-x-x^2 +x*O(x^n)))^(3*n+1)/(3*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=4 of A245049.

Sequence in context: A233832 A185756 A362364 * A124548 A324463 A139085

Adjacent sequences: A239104 A239105 A239106 * A239108 A239109 A239110

Keyword

nonn

Author

N. J. A. Sloane, Mar 26 2014

Status

approved