|
|
A000356
|
|
Number of rooted cubic maps with 2n nodes and a distinguished Hamiltonian cycle: (2n)!(2n+1)! / (n!^2*(n+1)!(n+2)!).
(Formerly M3978 N1647)
|
|
12
|
|
|
1, 5, 35, 294, 2772, 28314, 306735, 3476330, 40831076, 493684828, 6114096716, 77266057400, 993420738000, 12964140630900, 171393565105575, 2291968851019650, 30961684478686500, 422056646314726500
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
a(2n-1) is also the sum of the numbers of standard Young tableaux of size 2n+1 and of shapes (k+3,k+2,2^{n-2-k}), 0 <= k <= n-2. - Amitai Regev (amitai.regev(AT)weizmann.ac.il), Mar 10 2010
|
|
REFERENCES
|
Amitai Regev, Preprint. [From Amitai Regev (amitai.regev(AT)weizmann.ac.il), Mar 10 2010]
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
|
G.f.: (with offset 0) 3F2( [1, 3/2, 5/2], [3, 4], 16*x) = (1 - 2*x - 2F1( [-1/2, 1/2], [2], 16*x) ) / (4*x^2). - Olivier Gérard, Feb 16 2011
D-finite with recurrence (n+2)*(n+1)*a(n)-4*(2*n-1)*(2*n+1)*a(n-1)=0. - R. J. Mathar, Mar 03 2013
E.g.f.: (1/2)*(2F2(1/2,3/2; 2,3; 16*x) - 1).
a(n) ~ 2^(4*n+1)/(Pi*n^3). (End)
a(n) = Product_{1 <= i <= j <= n-1} (i + j + 3)/(i + j - 1).
a(n) = (2^(n-1)) * Product_{1 <= i <= j <= n-1} (i + j + 3)/(i + j) for n >= 1.
|
|
MAPLE
|
binomial(2*n, n)*binomial(2*n+1, n+1)/(n+1)/(n+2) ;
end proc:
|
|
MATHEMATICA
|
CoefficientList[ Series[1 + (HypergeometricPFQ[{1, 3/2, 5/2}, {3, 4}, 16 x] - 1), {x, 0, 17}], x]
Table[(2*n)!*(2*n+2)!/(2*n!*(n+1)!^2*(n+2)!), {n, 30}] (* Vincenzo Librandi, Mar 25 2012 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn,nice
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|