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 A219535 #29 Aug 10 2023 10:27:04
%S A219535 1,3,21,192,2001,22539,267276,3287496,41556585,536565225,7046232285,
%T A219535 93820316412,1263673602300,17186898452772,235709926636296,
%U A219535 3256050894487824,45263067114496665,632721425905230213,8888476706476318047,125418490224196533096,1776734673565844413929
%N A219535 G.f. satisfies A(x) = 1 + x*(2*A(x)^2 + A(x)^3).
%H A219535 Michael De Vlieger, <a href="/A219535/b219535.txt">Table of n, a(n) for n = 0..799</a>
%H A219535 Elżbieta Liszewska, Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.
%F A219535 Let G(x) = (1-2*x - sqrt(1 - 8*x + 4*x^2)) / (2*x), then g.f. A(x) satisfies:
%F A219535 (1) A(x) = (1/x)*Series_Reversion(x/G(x)),
%F A219535 (2) A(x) = G(x*A(x)) and G(x) = A(x/G(x)),
%F A219535 where x*G(x) is the g.f. of A047891.
%F A219535 Recurrence: 2*n*(2*n+1)*(11*n - 16)*a(n) = (649*n^3 - 1593*n^2 + 1130*n - 240)*a(n-1) + 16*(n-2)*(2*n-3)*(11*n-5)*a(n-2). - _Vaclav Kotesovec_, Dec 28 2013
%F A219535 a(n) ~ sqrt((33+17*sqrt(33))/11) * ((59+11*sqrt(33))/8)^n / (4 * sqrt(2*Pi) * n^(3/2)). - _Vaclav Kotesovec_, Dec 28 2013
%F A219535 From _Seiichi Manyama_, Jul 28 2020: (Start)
%F A219535 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n,k) * binomial(2*n+k+1,n)/(2*n+k+1).
%F A219535 a(n) = (1/(2*n+1)) * Sum_{k=0..n} 2^k * binomial(2*n+1,k) * binomial(3*n-k,n-k). (End)
%F A219535 From _Seiichi Manyama_, Aug 10 2023: (Start)
%F A219535 a(n) = (1/n) * Sum_{k=0..n-1} (-2)^k * 3^(n-k) * binomial(n,k) * binomial(3*n-k,n-1-k) for n > 0.
%F A219535 a(n) = (1/n) * Sum_{k=1..n} 3^k * binomial(n,k) * binomial(2*n,k-1) for n > 0. (End)
%e A219535 G.f.: A(x) = 1 + 3*x + 21*x^2 + 192*x^3 + 2001*x^4 + 22539*x^5 +...
%e A219535 Related expansions:
%e A219535 A(x)^2 = 1 + 6*x + 51*x^2 + 510*x^3 + 5595*x^4 + 65148*x^5 +...
%e A219535 A(x)^3 = 1 + 9*x + 90*x^2 + 981*x^3 + 11349*x^4 + 136980*x^5 +...
%e A219535 The g.f. satisfies A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where
%e A219535 G(x) = 1 + 3*x + 12*x^2 + 57*x^3 + 300*x^4 + 1686*x^5 +...+ A047891(n+1)*x^n +...
%t A219535 CoefficientList[1/x*InverseSeries[Series[2*x^2/(1-2*x-Sqrt[1-8*x+4*x^2]), {x, 0, 21}], x],x] (* _Vaclav Kotesovec_, Dec 28 2013 *)
%o A219535 (PARI) /* Formula A(x) = 1 + x*(2*A(x)^2 + A(x)^3): */
%o A219535 {a(n)=my(A=1);for(i=1,n,A=1+x*(2*A^2+A^3) +x*O(x^n));polcoeff(A,n)}
%o A219535 for(n=0,25,print1(a(n),", "))
%o A219535 (PARI) /* Formula using Series Reversion: */
%o A219535 {a(n)=my(A=1,G=(1-2*x-sqrt(1-8*x+4*x^2+x^3*O(x^n)))/(2*x));A=(1/x)*serreverse(x/G);polcoeff(A,n)}
%o A219535 for(n=0,25,print1(a(n),", "))
%o A219535 (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(n, k)*binomial(2*n+k+1, n)/(2*n+k+1)); \\ _Seiichi Manyama_, Jul 28 2020
%o A219535 (PARI) a(n) = sum(k=0, n, 2^k*binomial(2*n+1, k)*binomial(3*n-k, n-k))/(2*n+1); \\ _Seiichi Manyama_, Jul 28 2020
%Y A219535 Column k=2 of A336575.
%Y A219535 Cf. A047891, A219534, A219536.
%K A219535 nonn
%O A219535 0,2
%A A219535 _Paul D. Hanna_, Nov 21 2012