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 A215506 #5 Aug 13 2012 23:21:27
%S A215506 1,1,5,39,387,4516,59522,864981,13641761,230931591,4162594198,
%T A215506 79412758356,1595971048438,33662065586127,742852904575909,
%U A215506 17107642358401929,410242691108675077,10223836926997402818,264339609703677876178,7079673476173815446742
%N A215506 G.f. satisfies: A(x) = 1 + x*A(x)^4*A(x*A(x)^4).
%F A215506 a(n) = coefficient of x^n in (1+x*A(x))^(4*n+1)/(4*n+1) where A(x) = Sum_{n=0} a(n)*x^n.
%F A215506 Recurrence:
%F A215506 Let A(x)^m = Sum_{n>=0} a(n,m)*x^n with a(0,m)=1, then
%F A215506 a(n,m) = Sum_{k=0..n} m*binomial(4*n+m,k)/(4*n+m) * a(n-k,k).
%F A215506 G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n)^3*F(x,n+1)) for n>0 with F(x,0)=1.
%e A215506 G.f.: A(x) = 1 + x + 5*x^2 + 39*x^3 + 387*x^4 + 4516*x^5 + 59522*x^6 +...
%e A215506 G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:
%e A215506 A = 1 + x*A^3*B;
%e A215506 B = A*(1 + x*B^3*C);
%e A215506 C = B*(1 + x*C^3*D);
%e A215506 D = C*(1 + x*D^3*E);
%e A215506 E = D*(1 + x*E^3*F); ...
%e A215506 where B(x) = A(x)*A(x*A(x)^4), C(x) = A(x)*B(x*A(x)^4), D(x) = A(x)*C(x*A(x)^4), ...
%e A215506 Expansions of a few of the functions described above begin:
%e A215506 B(x) = 1 + 2*x + 15*x^2 + 158*x^3 + 1991*x^4 + 28283*x^5 + 439880*x^6 +...
%e A215506 C(x) = 1 + 3*x + 30*x^2 + 402*x^3 + 6257*x^4 + 107405*x^5 + 1983027*x^6 +...
%e A215506 D(x) = 1 + 4*x + 50*x^2 + 816*x^3 + 15215*x^4 + 308986*x^5 + 6679230*x^6 +...
%e A215506 E(x) = 1 + 5*x + 75*x^2 + 1445*x^3 + 31480*x^4 + 741075*x^5 + 18447355*x^6 +...
%e A215506 ALTERNATE GENERATING METHOD.
%e A215506 Suppose functions A=A(x), B=B(x), C=C(x), etc., satisfy:
%e A215506 A = 1 + x*A^4*B,
%e A215506 B = 1 + x*(A*B)^4*C,
%e A215506 C = 1 + x*(A*B*C)^4*D,
%e A215506 D = 1 + x*(A*B*C*D)^4*E, etc.,
%e A215506 then B(x) = A(x*A(x)^4), C(x) = B(x*A(x)^4), D(x) = C(x*A(x)^4), etc.,
%e A215506 where A(x) = 1 + x*A(x)^4*A(x*A(x)^4) is the g.f. of this sequence.
%e A215506 Expansions of a few of the functions described above begin:
%e A215506 B(x) = 1 + x + 9*x^2 + 105*x^3 + 1415*x^4 + 21089*x^5 + 340100*x^6 +...
%e A215506 C(x) = 1 + x + 13*x^2 + 203*x^3 + 3507*x^4 + 65018*x^5 + 1274266*x^6 +...
%e A215506 D(x) = 1 + x + 17*x^2 + 333*x^3 + 7047*x^4 + 157359*x^5 + 3665076*x^6 +...
%e A215506 E(x) = 1 + x + 21*x^2 + 495*x^3 + 12419*x^4 + 325312*x^5 + 8813506*x^6 +...
%o A215506 (PARI) {a(n)=local(A=1+x); for(i=0, n, A=1+x*A^4*subst(A, x, x*A^4+x*O(x^n))); polcoeff(A, n)}
%o A215506 (PARI) /* a(n) = [x^n] (1+x*A(x))^(4*n+1)/(4*n+1): */
%o A215506 {a(n)=local(A=1+x); for(i=0, n, A=sum(m=0,n,polcoeff((1+x*A+x*O(x^m))^(4*m+1)/(4*m+1),m)*x^m)+x*O(x^n));polcoeff(A,n)}
%o A215506 (PARI) {a(n, m=1)=if(n==0, 1, if(m==0, 0^n, sum(k=0, n, m*binomial(4*n+m, k)/(4*n+m)*a(n-k, k))))}
%o A215506 for(n=0,41,print1(a(n),", "))
%Y A215506 Cf. A088717, A215505, A215507.
%K A215506 nonn
%O A215506 0,3
%A A215506 _Paul D. Hanna_, Aug 13 2012