Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #28 May 16 2023 12:21:10
%S 1,2,19,253,3920,66221,1183077,21981764,420449439,8223704755,
%T 163727846678,3307039145618,67600147666909,1395822347989531,
%U 29070233296701815,609950649080323320,12881240945694949696,273590092192962485985,5840400740191969187922
%N Number of hybrid 5-ary trees with n internal nodes.
%H Alois P. Heinz, <a href="/A239108/b239108.txt">Table of n, a(n) for n = 0..300</a>
%H SeoungJi Hong and SeungKyung Park, <a href="http://dx.doi.org/10.4134/BKMS.2014.51.1.229">Hybrid d-ary trees and their generalization</a>, Bull. Korean Math. Soc. 51 (2014), No. 1, pp. 229-235. See p. 233.
%H Sheng-liang Yang and Mei-yang Jiang, <a href="https://journal.lut.edu.cn/EN/abstract/abstract528.shtml">Pattern avoiding problems on the hybrid d-trees</a>, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. (in Mandarin)
%F From _Paul D. Hanna_, Mar 30 2014: (Start)
%F G.f. A(x) satisfies:
%F (1) A(x) = (1 + x*A(x)^4) * (1 + x*A(x)^5).
%F (2) A(x) = ( (1/x)*Series_Reversion( x*(1-x-x^2)^4/(1+x)^4 ) )^(1/4).
%F (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 ).
%F (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 ).
%F (5) A(x) = Sum_{n>=0} Fibonacci(n+2) * x^n * A(x)^(4*n).
%F (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).
%F The formal inverse of g.f. A(x) is (sqrt(1-2*x+5*x^2) - (1+x))/(2*x^5).
%F a(n) = [x^n] ( (1+x)/(1-x-x^2) )^(4*n+1) / (4*n+1).
%F (End)
%t (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 *)
%o (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)
%o for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Mar 30 2014
%o (PARI) a(n)=polcoeff( ((1/x)*serreverse( x*(1-x-x^2)^4/(1+x +x*O(x^n))^4))^(1/4), n)
%o for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Mar 30 2014
%o (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)
%o for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Mar 30 2014
%o (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)
%o for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Mar 30 2014
%o (PARI) a(n)=polcoeff(((1+x)/(1-x-x^2 +x*O(x^n)))^(4*n+1)/(4*n+1), n)
%o for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Mar 30 2014
%Y Cf. A000045, A007863, A215654, A239107, A239108, A239109.
%Y Column k=5 of A245049.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Mar 26 2014