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 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 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: A248053 A002294 A051406 * A027392 A291813 A346765 Adjacent sequences: A000353 A000354 A000355 * A000357 A000358 A000359 KEYWORD easy,nonn,nice AUTHOR EXTENSIONS Better definition from Michael Albert, Oct 24 2008 STATUS approved

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Last modified March 28 03:48 EDT 2023. Contains 361577 sequences. (Running on oeis4.)