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%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
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