OFFSET
0,2
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Gheorghe Coserea, Table of n, a(n) for n = 0..200
Dov Tamari, Monoïdes préordonnés et chaînes de Malcev, Bulletin de la Société Mathématique de France, Volume 82 (1954), 53-96. See end of Appendix II.
T. R. S. Walsh, A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259. See Table IVc.
FORMULA
From Paul D. Hanna, Nov 26 2009: (Start)
G.f.: A(x) = [x/Series_Reversion(x*F(x)^2)]^(1/2) where F(x) = g.f. of A005568, where A005568(n) is the product of two successive Catalan numbers C(n)*C(n+1).
G.f.: A(x) = F(x/A(x)^2) where A(x*F(x)^2) = F(x) where F(x) = g.f. of A005568.
G.f.: A(x) = G(x/A(x)) where A(x*G(x)) = G(x) where F(x) = g.f. of A168450.
G.f.: A(x) = x/Series_Reversion(x*G(x)) where G(x) = g.f. of A168450.
Self-convolution yields A168451.
(End)
MAPLE
A004304 := proc(n) local N, x, ode ; Order := n+1 ; ode := x^2*diff(N(x), x, x)*(N(x)^3-16*x*N(x)) ; ode := ode + (x*diff(N(x), x))^3*(16-6*N(x)) ; ode := ode + (x*diff(N(x), x))^2*(12*N(x)^2-16*x-24*N(x)) ; ode := ode + x*diff(N(x), x)*(-8*N(x)^3+24*x*N(x)+12*N(x)^2) ; ode := ode + 2*N(x)^2*(N(x)^2-N(x)-6*x) ; dsolve({ode=0, N(0)=1, D(N)(0)=2}, N(x), type=series) ; convert(%, polynom) ; rhs(%) ; RETURN( coeftayl(%, x=0, n)) ; end; for n from 0 to 20 do printf("%d, ", A004304(n)) ; od ; # R. J. Mathar, Aug 18 2006
MATHEMATICA
m = 22;
F[x_] = Sum[2 (2n+1) Binomial[2n, n]^2 x^n/((n+2)(n+1)^2), {n, 0, m}];
A[x_] = (x/InverseSeries[x F[x]^2 + O[x]^m, x])^(1/2);
CoefficientList[A[x], x] (* Jean-François Alcover, Mar 28 2020 *)
PROG
(PARI) {a(n)=local(C_2=vector(n+1, m, (binomial(2*m-2, m-1)/m)*(binomial(2*m, m)/(m+1)))); polcoeff((x/serreverse(x*Ser(C_2)^2))^(1/2), n)} \\ Paul D. Hanna, Nov 26 2009
(PARI)
seq(N) = {
my(c(n)=binomial(2*n, n)/(n+1), s=Ser(apply(n->c(n)*c(n+1), [0..N])));
Vec(subst(s, 'x, serreverse('x*s^2)));
};
seq(20)
\\ test: y=Ser(seq(200)); 0 == x^2*y''*(y^3 - 16*x*y) + (x*y')^3*(16-6*y) + (x*y')^2*(12*y^2-16*x-24*y) + x*y'*(-8*y^3 + 24*x*y + 12*y^2) + 2*y^2*(y^2-y-6*x)
\\ Gheorghe Coserea, Jun 13 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from R. J. Mathar, Aug 18 2006
STATUS
approved