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.
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)^4) * (1 + x*A(x)^5).
(2) A(x) = ( (1/x)*Series_Reversion( x*(1-x-x^2)^4/(1+x)^4 ) )^(1/4).
(3) A(x) = exp( Sum_{n>=1} x^n*A(x)^(3*n)/n * Sum_{k=0..n} C(n,k)^2 * A(x)^k ).
(4) A(x) = exp( Sum_{n>=1} x^n*A(x)^(4*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)^(4*n).
(6) A(x) = G(x*A(x)^3) where G(x) = A(x/G(x)^3) 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^5).
a(n) = [x^n] ( (1+x)/(1-x-x^2) )^(4*n+1) / (4*n+1).
(End)
MATHEMATICA
(1/x InverseSeries[x(1 - x - x^2)^4/(1 + x)^4 + O[x]^20])^(1/4) // 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^4)*(1 + x*A^5)); 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)^4/(1+x +x*O(x^n))^4))^(1/4), 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)
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)=polcoeff(((1+x)/(1-x-x^2 +x*O(x^n)))^(4*n+1)/(4*n+1), n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 30 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Mar 26 2014
STATUS
approved