OFFSET
0,3
COMMENTS
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
LINKS
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962, 2014.
FORMULA
EXAMPLE
G.f.: A(x) = F(x*G(x)) = 1 + x + 3*x^2 + 13*x^3 + 69*x^4 +... where
F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
F(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 +...
G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 +...
G(x)^4 = 1 + 4*x + 22*x^2 + 140*x^3 + 969*x^4 + 7084*x^5 +...
A(x)^2 = 1 + 2*x + 7*x^2 + 32*x^3 + 173*x^4 + 1050*x^5 +...
G(x)*A(x)^2 = 1 + 3*x + 13*x^2 + 69*x^3 + 417*x^4 + 2754*x^5 +...
MATHEMATICA
nmax = 21;
G[_] = 0;
Do[G[x_] = 1 + x*G[x]^4 + O[x]^nmax, nmax];
F[x_] = Sum[CatalanNumber[n] x^n, {n, 0, nmax}];
A[x_] = F[x G[x]];
CoefficientList[A[x], x] (* Jean-François Alcover, Sep 09 2018 *)
PROG
(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)))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 15 2009
STATUS
approved