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A000356 Number of rooted cubic maps with 2n nodes and a distinguished Hamilton cycle: (2n)!(2n+1)! / (n!^2*(n+1)!(n+2)!).
(Formerly M3978 N1647)
11
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

Vincenzo Librandi, Table of n, a(n) for n = 1..800

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6

W. T. Tutte, A census of Hamiltonian polygons, Canad. J. Math., 14 (1962), 402-417.

W. T. Tutte, On the enumeration of four-colored maps, SIAM J. Appl. Math., 17 (1969), 454-460.

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

a(n)*(n+2) = A000891(n). - Gary W. Adamson, Apr 08 2011

(n+2)*(n+1)*a(n)-4*(2*n-1)*(2*n+1)*a(n-1)=0. - R. J. Mathar, Mar 03 2013

From Ilya Gutkovskiy, Feb 01 2017: (Start)

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)

MAPLE

A000356 := proc(n)

    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

Cf. A000264, A000309.

Equals A005568/2.

Fourth row of array A102539.

Column of array A073165.

Image of A001700 under the "little Hankel" transform (see A056220 for definition). - John W. Layman, Aug 22 2000

Cf. A000891.

Sequence in context: A248053 A002294 A051406 * A027392 A291813 A177354

Adjacent sequences:  A000353 A000354 A000355 * A000357 A000358 A000359

KEYWORD

easy,nonn,nice

AUTHOR

N. J. A. Sloane, Simon Plouffe

EXTENSIONS

Better definition from Michael Albert, Oct 24 2008

STATUS

approved

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Last modified September 20 16:09 EDT 2017. Contains 292276 sequences.