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A213282 G.f. satisfies A(x) = G(x/(1-x)^3) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

%I A213282 #29 Oct 03 2023 08:59:27
%S A213282 1,1,6,36,236,1656,12192,92960,727824,5817696,47281472,389533056,
%T A213282 3245867136,27308274688,231654031104,1979205694464,17016094611712,
%U A213282 147104972637696,1277988764697600,11151534242977792,97692088569096192,858890594909048832,7575804347863105536
%N A213282 G.f. satisfies A(x) = G(x/(1-x)^3) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.
%C A213282 Compare to the g.f. B(x) of A006319 where B(x) = C(x/(1-x)^2) such that C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
%H A213282 Vincenzo Librandi, <a href="/A213282/b213282.txt">Table of n, a(n) for n = 0..200</a>
%F A213282 G.f. satisfies: A(x) = F(x*A(x)^3) where F(x) = 1 + x/F(-x)^3 is the g.f. of A213281.
%F A213282 G.f. A(x) satisfies: A(1 - G(-x)) = G(x) = 1 + x*G(x)^3 is the g.f. of A001764.
%F A213282 a(n) = Sum_{k=0..n} binomial(n+2*k-1,n-k) * binomial(3*k,k)/(2*k+1). - _Seiichi Manyama_, Oct 03 2023
%e A213282 G.f.: A(x) = 1 + x + 6*x^2 + 36*x^3 + 236*x^4 + 1656*x^5 + 12192*x^6 +...
%e A213282 G.f.: A(x) = G(x/(1-x)^3) where G(x) = 1 + x*G(x)^3 is g.f. of A001764:
%e A213282 G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
%p A213282 series(RootOf(G = 1 + G^3*x/(1-x)^3, G),x=0,30); # _Mark van Hoeij_, Apr 18 2013
%o A213282 (PARI) /* G.f. A(x) = G(x/(1-x)^3) where G(x) = 1 + x*G(x)^3: */
%o A213282 {a(n)=local(A,G=1+x);for(i=1,n,G=1+x*G^3+x*O(x^n));A=subst(G,x,x/(1-x+x*O(x^n))^3);polcoeff(A,n)}
%o A213282 for(n=0,25,print1(a(n),", "))
%o A213282 (PARI) /* G.f. A(x) = F(x*A(x)^3) where F(x) = 1 + x/F(-x)^3: */
%o A213282 {a(n)=local(F=1+x+x*O(x^n), A=1); for(i=1, n+1, F=1+x/subst(F^3, x, -x+x*O(x^n))); A=(serreverse(x/F^3)/x)^(1/3); polcoeff(A, n)}
%o A213282 for(n=0, 25, print1(a(n), ", "))
%Y A213282 Cf. A213281, A001764; variants: A006319 (royal paths in a lattice), A213336.
%K A213282 nonn
%O A213282 0,3
%A A213282 _Paul D. Hanna_, Jun 08 2012