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)

1, 5, 35, 294, 2772, 28314, 306735, 3476330, 40831076, 493684828, 6114096716, 77266057400, 993420738000, 12964140630900, 171393565105575, 2291968851019650, 30961684478686500, 422056646314726500

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.

Anatol N. Kirillov, Notes on Schubert, Grothendieck and key polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 034, 56 p. (2016).

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

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

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)

From Peter Bala, Feb 22 2023: (Start)

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.

Cf. A003645. (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: A002294 A369128 A051406 * A370844 A379103 A027392

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