|
| |
|
|
A004304
|
|
Number of planar tree-rooted maps with n edges.
(Formerly M0364)
|
|
6
| |
|
|
1, 2, 2, 6, 28, 160, 1036, 7294, 54548, 426960, 3463304, 28910816, 247104976, 2154192248, 19097610480, 171769942086, 1564484503044, 14407366963440, 133978878618904, 1256799271555872, 11881860129979440
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
REFERENCES
| N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Walsh, T. R. S.; Lehman, A. B.; Counting rooted maps by genus. III: Nonseparable maps. J. Combinatorial Theory Ser. B 18 (1975), 222-259.
|
|
|
FORMULA
| Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), 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 (mathar(AT)strw.leidenuniv.nl), Aug 18 2006
|
|
|
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)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 26 2009]
|
|
|
CROSSREFS
| Cf. A000264.
Cf. A005568, A168450, A168451, A168452. [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 26 2009]
Sequence in context: A076726 A032272 A179320 * A108800 A058250 A179929
Adjacent sequences: A004301 A004302 A004303 * A004305 A004306 A004307
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 18 2006
|
| |
|
|
