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 A179486 #16 May 14 2012 22:57:23
%S A179486 1,1,3,12,56,285,1533,8571,49311,290019,1735845,10538550,64741482,
%T A179486 401708636,2513837931,15847467276,100547976684,641571037002,
%U A179486 4114313992851,26503239829953,171416342026944,1112726829455289
%N A179486 G.f. A(x) satisfies A(x) = x + A(A(x)^3) where A(x) = Sum_{n>=1} a(n)*x^(2*n-1).
%F A179486 G.f. A(x) satisfies: A(x - A(x^3)) = x.
%F A179486 G.f.: A(x) = x + G(x) + G(G(x)) + G(G(G(x))) + G(G(G(G(x)))) +... where G(x) = A(x)^3 = g.f. of A179487.
%F A179486 Given g.f. A(x), define C(x) = A(x^3), then C(x) = x^3 + C(C(x)).
%F A179486 Derivative of g.f. A(x) satisfies: A'(x) = 1/(1 - 3*A(x)^2*A'(A(x)^3)).
%F A179486 Radius of convergence, r, and related values:
%F A179486 r = 0.378590868760339249702289974755919481665219504207137681721365...;
%F A179486 A(r) = 0.5510035125320818261355419567786182869427265480378585343298... where r = A(r) - A(A(r)^3);
%F A179486 A(r)^3 = 0.1672873502451522851544780724841939477291722823741494215...;
%F A179486 A(A(r)^3) = 0.1724126437717425764332519820226988052775070438307208...;
%F A179486 A'(A(r)^3) = 1.0979182660346808662695442970765885990300854399844658... where A'(A(r)^3) = 1/(3*A(r)^2);
%F A179486 Limit a(n+1)/a(n) = 1/r^2 = 6.9768555281242291444841704586123374638...
%F A179486 Let V(x) = x/(x - A(x^3)) then V'(A(r)) = 1/r, V(z) - z*V'(z) = 0 at z=A(r), and V(A(x)) = A(x)/x for all x.
%e A179486 G.f.: A(x) = x + x^3 + 3*x^5 + 12*x^7 + 56*x^9 + 285*x^11 +...
%e A179486 A(x)^3 = x^3 + 3*x^5 + 12*x^7 + 55*x^9 + 276*x^11 + 1470*x^13 +...
%e A179486 The series reversion of A(x) equals x - A(x^3), therefore
%e A179486 x = A(x - x^3 - x^9 - 3*x^15 - 12*x^21 - 56*x^27 - 285*x^33 -...).
%e A179486 Let G(x) = A(x)^3 be the g.f. of A179487, then
%e A179486 G(G(x)) = x^9 + 9*x^11 + 63*x^13 + 411*x^15 + 2619*x^17 +...,
%e A179486 G(G(G(x))) = x^27 + 27*x^29 + 432*x^31 + 5364*x^33 +..., and
%e A179486 G(G(G(G(x)))) = x^81 + 81*x^83 + 3483*x^85 +...
%e A179486 where A(x) = x + G(x) + G(G(x)) + G(G(G(x))) + G(G(G(G(x)))) +...
%o A179486 (PARI) {a(n)=local(A=x+x^3); for(i=0, n, A=serreverse(x-subst(A, x, x^3+x^2*O(x^(2*n))))); polcoeff(A, 2*n-1)}
%o A179486 (Maxima)
%o A179486 Co(n,k,F):=if k=1 then F(n) else sum(F(i+1)*Co(n-i-1, k-1, F),i,0,n-k);
%o A179486 b(n):=if n=1 then 1 else sum(if 3*k>n then 0 else Co(n,3*k,b)*b(k),k,1, n);
%o A179486 a(n):=b(2*n-1);
%o A179486 makelist(a(n),n,1,7); [_Vladimir Kruchinin_, Jun 28 2011]
%o A179486 (Maxima)
%o A179486 T(n,m):=if n=m then 1 else kron_delta(n,m)+sum(binomial(m,j)*sum(if 3*k<=n-j then T(n-j,3*k)*T(k,m-j) else 0,k,m-j,n-j),j,0,m-1);
%o A179486 makelist(T(n,1),n,1,12); [_Vladimir Kruchinin_, May 02 2012]
%Y A179486 Cf. A179487, A141200 (variant).
%K A179486 nonn
%O A179486 1,3
%A A179486 _Paul D. Hanna_, Aug 12 2010
%E A179486 Typo in example corrected by _Paul D. Hanna_, Aug 13 2010