|
|
A000257
|
|
Number of rooted bicubic maps: a(n) = (8*n-4)*a(n-1)/(n+2) for n >= 2, a(0) = a(1) = 1.
(Formerly M2927 N1175)
|
|
25
|
|
|
1, 1, 3, 12, 56, 288, 1584, 9152, 54912, 339456, 2149888, 13891584, 91287552, 608583680, 4107939840, 28030648320, 193100021760, 1341536993280, 9390758952960, 66182491668480, 469294031831040, 3346270487838720, 23981605162844160, 172667557172477952
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
Number of indecomposable 1342-avoiding permutations of length n.
a(n) is also the number of rooted planar hypermaps with n darts (darts are semi-edges in the particular case of ordinary maps). - Valery A. Liskovets, Apr 13 2006
Number of "new" intervals in Tamari lattices of size n (see Chapoton paper). - Ralf Stephan, May 08 2007
|
|
REFERENCES
|
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 321.
L. M. Koganov, V. A. Liskovets, T. R. S. Walsh, Total vertex enumeration in rooted planar maps, Ars Combin., Vol. 54 (2000), pp. 149-160.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
LINKS
|
|
|
FORMULA
|
a(0) = 1 and a(n) = 3*2^(n-1)*C(n)/(n+2) for n >= 1, where C = Catalan (A000108).
a(n) = 2^(n-2) * A007054(n), n > 1.
O.g.f.: 1/4 + (1/8) * ( -(1-8*x)^(1/2) + 16*(1-8*x)^(1/2)*x+1-8*x ) / ((1-8*x)^(1/2)*x*(1+(1-8*x)^(1/2))). - Karol A. Penson, Jun 04 2004
E.g.f.: (1/8) * exp(4*x)*(8*BesselI(0, 4*x)*x-BesselI(1, 4*x)-8*BesselI(1, 4*x)*x)/x. - Karol A. Penson, Jun 04 2004
D-finite with recurrence (n + 2) * a(n) = (8*n - 4) * a(n - 1). - Simon Plouffe, Feb 09 2012
O.g.f.: ((1-8*x)^(3/2) + 8*x^2 + 12*x - 1)/(32*x^2) = 1 + x + 3*x^2 + 12*x^3 + 56*x^4 + .... The related generating function 1 + 3*x^2 + 12*x^4 + 56*x^6 + ... is the zeta function associated to a certain 2 X 2 matrix of noncommuting variables. See Kassel and Reutenauer, Example 5.1. - Peter Bala, Mar 15 2013
0 = a(n) * (64*a(n+1) - 28*a(n+2)) + a(n+1) * (12*a(n+1) + a(n+2)) if n > 0. - Michael Somos, Apr 03 2014
Integral representation as the n-th moment of the positive function W(x) on (0,8). a(n) = Integral_{x=0..8} x^n*W(x) dx, n=1,2,3,..., where W(x) = sqrt((8-x)^3/x)/(32*Pi). For n=0 the integral is equal to 3/4. This means that a(n) is the n-th moment, n=0,1,2,..., of the probability distribution which is a sum of W(x) as the continuous part and an atom at x=0 with weight 1/4 (Dirac(x)/4). This representation is unique as W(x) is the solution of the Hausdorff moment problem. - Karol A. Penson and Wojciech Mlotkowski, Jul 15 2015
G.f. y satisfies: 0 = 16*x^2*y^2 - (8*x^2+12*x-1)*y + x^2+11*x-1. - Gheorghe Coserea, Nov 22 2016
Sum_{n>=0} 1/a(n) = 1985/1029 + 1280*arcsin(1/(2*sqrt(2)))/(343*sqrt(7)).
Sum_{n>=0} (-1)^n/a(n) = 341/729 - 1280*arcsinh(1/(2*sqrt(2)))/2187. (End)
|
|
EXAMPLE
|
G.f. = 1 + x + 3*x^2 + 12*x^3 + 56*x^4 + 288*x^5 + 1584*x^6 + 9152*x^7 + ...
|
|
MAPLE
|
option remember;
if n <=1 then
1;
else
4*(2*n-1)*procname(n-1)/(n+2) ;
end if ;
|
|
MATHEMATICA
|
CoefficientList[Series[1 + x HypergeometricPFQ[{1, 3/2}, {4}, 8 x], {x, 0, 10}], x]
(* Second program: *)
Join[{1}, Table[3*2^(n-1) CatalanNumber[n]/(n+2), {n, 30}]] (* Harvey P. Dale, Dec 18 2011 *)
|
|
PROG
|
(PARI)
C(n)=binomial(2*n, n)/(n+1);
a(n)=if(n==0, 1, 3*2^(n-1)*C(n)/(n+2) ); \\ Joerg Arndt, May 04 2013
(PARI) x='x+O('x^66); Vec( ((1-8*x)^(3/2) + 8*x^2 + 12*x - 1)/(32*x^2) ) \\ Joerg Arndt, May 04 2013
(PARI)
x='x; y='y; Fxy = 16*x^2*y^2 - (8*x^2+12*x-1)*y + x^2+11*x-1;
seq(N) = {
my(y0 = 1 + O('x^N), y1=0);
for (k = 1, N,
y1 = y0 - subst(Fxy, y, y0)/subst(deriv(Fxy, y), y, y0);
if (y1 == y0, break()); y0 = y1);
Vec(y0);
};
(Magma) [1] cat [3*2^n*Factorial(2*n)/((2*n^2+6*n+4)*Factorial(n)^2): n in [1.. 25]]; // Vincenzo Librandi, Oct 21 2014
(Python)
a000257 = [1]
for n in range(1, 25): a000257.append((8*n-4)*a000257[-1]//(n+2))
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|