A004304 Number of nonseparable planar tree-rooted maps with n edges. (Formerly M0364)

1, 2, 2, 6, 28, 160, 1036, 7294, 54548, 426960, 3463304, 28910816, 247104976, 2154192248, 19097610480, 171769942086, 1564484503044, 14407366963440, 133978878618904, 1256799271555872, 11881860129979440

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

Cf. A000264.

Cf. A005568, A168450, A168451, A168452. - Paul D. Hanna, Nov 26 2009

Sequence in context: A032272 A214446 A179320 * A326907 A270487 A058250

Adjacent sequences: A004301 A004302 A004303 * A004305 A004306 A004307

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from R. J. Mathar, Aug 18 2006

Status

approved