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 A219534 #29 Apr 03 2024 11:12:16
%S A219534 1,2,12,100,968,10208,113792,1318832,15732064,191878592,2381917824,
%T A219534 29995598208,382257383168,4920505410816,63882881030656,
%U A219534 835554927932160,10999486798112256,145626782310460416,1937772463214168064,25901381584638605312,347618773649248088064
%N A219534 G.f. satisfies A(x) = 1 + x*(A(x)^2 + A(x)^4).
%H A219534 G. C. Greubel, <a href="/A219534/b219534.txt">Table of n, a(n) for n = 0..850</a>
%F A219534 Let G(x) = (1 - sqrt(1-4*x-4*x^2))/(2*x), then g.f. A(x) satisfies:
%F A219534 (1) A(x) = sqrt( (1/x)*Series_Reversion(x/G(x)^2) ),
%F A219534 (2) A(x) = G(x*A(x)^2) and G(x) = A(x/G(x)^2),
%F A219534 where x*G(x) is the g.f. of A025227.
%F A219534 Recurrence: 3*n*(3*n-1)*(3*n+1)*(131*n^3 - 666*n^2 + 1075*n - 558)*a(n) = 2*(26200*n^6 - 172500*n^5 + 431572*n^4 - 521613*n^3 + 316327*n^2 - 89058*n + 8640)*a(n-1) - 12*(n-2)*(1441*n^5 - 8767*n^4 + 19186*n^3 - 18172*n^2 + 6930*n - 810)*a(n-2) + 8*(n-3)*(n-2)*(2*n-5)*(131*n^3 - 273*n^2 + 136*n - 18)*a(n-3). - _Vaclav Kotesovec_, Sep 10 2013
%F A219534 a(n) ~ c*d^n/n^(3/2), where d = 2/81*(7217783 + 10611 * sqrt(786))^(1/3) + 74654/(81*(7217783 + 10611 * sqrt(786))^(1/3)) + 400/81 = 14.48001092254652246... is the root of the equation -16 + 132*d - 400*d^2 + 27*d^3 = 0 and c = 1/2358*sqrt(262)*sqrt((213070976 + 3034746 * sqrt(786))^(1/3) * ((213070976 + 3034746 * sqrt(786))^(2/3) + 336670 + 1310*(213070976 + 3034746 * sqrt(786))^(1/3)))/((213070976 + 3034746 * sqrt(786))^(1/3)*sqrt(Pi)) = 0.1929450901182412149... - _Vaclav Kotesovec_, Sep 10 2013
%F A219534 a(n) = (1/n) * Sum_{k=0..floor(n-1)/2} 2^(n-k) * binomial(n,k) * binomial(3*n-k,n-1-2*k) for n > 0. - _Seiichi Manyama_, Apr 01 2024
%F A219534 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(2*n+2*k+1,n)/(2*n+2*k+1). - _Seiichi Manyama_, Apr 03 2024
%e A219534 G.f.: A(x) = 1 + 2*x + 12*x^2 + 100*x^3 + 968*x^4 + 10208*x^5 +...
%e A219534 Related expansions:
%e A219534 A(x)^2 = 1 + 4*x + 28*x^2 + 248*x^3 + 2480*x^4 + 26688*x^5 +...
%e A219534 A(x)^4 = 1 + 8*x + 72*x^2 + 720*x^3 + 7728*x^4 + 87104*x^5 +...
%e A219534 The g.f. satisfies A(x) = G(x*A(x)^2) and G(x) = A(x/G(x)^2) where
%e A219534 G(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 40*x^4 + 144*x^5 + 544*x^6 +...+ A025227(n+1)*x^n +...
%t A219534 nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=2; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[1+x*(AGF^2+AGF^4)-AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* _Vaclav Kotesovec_, Sep 10 2013 *)
%o A219534 (PARI) /* Formula A(x) = 1 + x*(A(x)^2 + A(x)^4): */
%o A219534 {a(n)=local(A=1);for(i=1,n,A=1+x*(A^2+A^4) +x*O(x^n));polcoeff(A,n)}
%o A219534 for(n=0,25,print1(a(n),", "))
%o A219534 (PARI) /* Formula using Series Reversion: */
%o A219534 {a(n)=local(A=1,G=(1-sqrt(1-4*x-4*x^2+x^3*O(x^n)))/(2*x));A=sqrt((1/x)*serreverse(x/G^2));polcoeff(A,n)}
%o A219534 for(n=0,25,print1(a(n),", "))
%Y A219534 Cf. A025227, A219535, A219536, A219537, A219538.
%Y A219534 Cf. A006318, A364167.
%K A219534 nonn
%O A219534 0,2
%A A219534 _Paul D. Hanna_, Nov 21 2012