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%I #9 Sep 09 2018 08:25:10
%S 1,1,3,13,69,417,2754,19373,142732,1088875,8533278,68308641,556242792,
%T 4593529882,38380159009,323860968709,2756019889146,23625552635184,
%U 203823793118268,1768357487401595,15418860927887232,135042445950316514
%N G.f.: A(x) = F(x*G(x)) where F(x) = G(x/F(x)^2) = 1 + x*F(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(x*G(x)^2) = 1 + x*G(x)^4 is the g.f. of A002293.
%C This appears to be the same as the sequence in row 1 of Fig. 21 of Novelli-Thibon 2014. - _N. J. A. Sloane_, Jun 14 2014
%H J.-C. Novelli, J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962, 2014.
%F a(n) = Sum_{k=0..n} C(2k+1,k)/(2k+1) * C(4n-3k,n-k)*k/(4n-3k) for n>0 with a(0)=1.
%F G.f. satisfies: A(x) = 1 + x*G(x)*A(x)^2 where G(x) is the g.f. of A002293.
%F G.f. satisfies: A(x/F(x)^2) = F(x/F(x)) where F(x) is the g.f. of A000108.
%F G.f. satisfies: A(x/H(x)) = F(x) where H(x) = 1 + x*H(x)^3 is the g.f. of A001764 and F(x) is the g.f. of A000108.
%e G.f.: A(x) = F(x*G(x)) = 1 + x + 3*x^2 + 13*x^3 + 69*x^4 +... where
%e F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
%e F(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
%e G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
%e G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 +...
%e G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 +...
%e G(x)^4 = 1 + 4*x + 22*x^2 + 140*x^3 + 969*x^4 + 7084*x^5 +...
%e A(x)^2 = 1 + 2*x + 7*x^2 + 32*x^3 + 173*x^4 + 1050*x^5 +...
%e G(x)*A(x)^2 = 1 + 3*x + 13*x^2 + 69*x^3 + 417*x^4 + 2754*x^5 +...
%t nmax = 21;
%t G[_] = 0;
%t Do[G[x_] = 1 + x*G[x]^4 + O[x]^nmax, nmax];
%t F[x_] = Sum[CatalanNumber[n] x^n, {n, 0, nmax}];
%t A[x_] = F[x G[x]];
%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Sep 09 2018 *)
%o (PARI) {a(n)=if(n==0,1,sum(k=0,n,binomial(2*k+1,k)/(2*k+1)*binomial(4*(n-k)+k,n-k)*k/(4*(n-k)+k)))}
%Y Cf. A000108, A001764, A002293; A153394, A153396.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 15 2009
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