This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A239107 #30 May 16 2023 12:21:15
%S A239107 1,2,15,155,1854,24124,331575,4736345,69616637,1046054129,15995716263,
%T A239107 248111418112,3894303176880,61737213540306,987116931080661,
%U A239107 15899835212249761,257758369219909534,4202381519278498915,68859442092723799788,1133401910867109123200
%N A239107 Number of hybrid 4-ary trees with n internal nodes.
%H A239107 Alois P. Heinz, <a href="/A239107/b239107.txt">Table of n, a(n) for n = 0..800</a>
%H A239107 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 A239107 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 A239107 From _Paul D. Hanna_, Mar 30 2014: (Start)
%F A239107 G.f. A(x) satisfies:
%F A239107 (1) A(x) = (1 + x*A(x)^3) * (1 + x*A(x)^4).
%F A239107 (2) A(x) = ( (1/x)*Series_Reversion( x*(1-x-x^2)^3/(1+x)^3 ) )^(1/3).
%F A239107 (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 ).
%F A239107 (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 ).
%F A239107 (5) A(x) = Sum_{n>=0} Fibonacci(n+2) * x^n * A(x)^(3*n).
%F A239107 (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).
%F A239107 The formal inverse of g.f. A(x) is (sqrt(1-2*x+5*x^2) - (1+x))/(2*x^4).
%F A239107 a(n) = [x^n] ( (1+x)/(1-x-x^2) )^(3*n+1) / (3*n+1).
%F A239107 (End)
%F A239107 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
%t A239107 (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 *)
%o A239107 (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)
%o A239107 for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Mar 30 2014
%o A239107 (PARI) a(n)=polcoeff( ((1/x)*serreverse( x*(1-x-x^2)^3/(1+x +x*O(x^n))^3))^(1/3), n)
%o A239107 for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Mar 30 2014
%o A239107 (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)
%o A239107 for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Mar 30 2014
%o A239107 (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
%o A239107 for(n=0, 20, print1(a(n), ", "))
%o A239107 (PARI) a(n)=polcoeff(((1+x)/(1-x-x^2 +x*O(x^n)))^(3*n+1)/(3*n+1), n)
%o A239107 for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Mar 30 2014
%Y A239107 Cf. A000045, A007863, A215654, A239107, A239108, A239109.
%Y A239107 Column k=4 of A245049.
%K A239107 nonn
%O A239107 0,2
%A A239107 _N. J. A. Sloane_, Mar 26 2014