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%I M3978 N1647 #68 Feb 24 2023 02:05:46
%S 1,5,35,294,2772,28314,306735,3476330,40831076,493684828,6114096716,
%T 77266057400,993420738000,12964140630900,171393565105575,
%U 2291968851019650,30961684478686500,422056646314726500
%N Number of rooted cubic maps with 2n nodes and a distinguished Hamiltonian cycle: (2n)!(2n+1)! / (n!^2*(n+1)!(n+2)!).
%C 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
%D Amitai Regev, Preprint. [From Amitai Regev (amitai.regev(AT)weizmann.ac.il), Mar 10 2010]
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A000356/b000356.txt">Table of n, a(n) for n = 1..800</a>
%H R. K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
%H Anatol N. Kirillov, <a href="https://doi.org/10.3842/SIGMA.2016.034">Notes on Schubert, Grothendieck and key polynomials</a>, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 034, 56 p. (2016).
%H W. T. Tutte, <a href="https://doi.org/10.4153/CJM-1962-032-x">A census of Hamiltonian polygons</a>, Canad. J. Math., 14 (1962), 402-417.
%H W. T. Tutte, <a href="http://dx.doi.org/10.1137/0117044">On the enumeration of four-colored maps</a>, SIAM J. Appl. Math., 17 (1969), 454-460.
%F 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
%F a(n)*(n+2) = A000891(n). - _Gary W. Adamson_, Apr 08 2011
%F 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
%F From _Ilya Gutkovskiy_, Feb 01 2017: (Start)
%F E.g.f.: (1/2)*(2F2(1/2,3/2; 2,3; 16*x) - 1).
%F a(n) ~ 2^(4*n+1)/(Pi*n^3). (End)
%F From _Peter Bala_, Feb 22 2023: (Start)
%F a(n) = Product_{1 <= i <= j <= n-1} (i + j + 3)/(i + j - 1).
%F a(n) = (2^(n-1)) * Product_{1 <= i <= j <= n-1} (i + j + 3)/(i + j) for n >= 1.
%F Cf. A003645. (End)
%p A000356 := proc(n)
%p binomial(2*n,n)*binomial(2*n+1,n+1)/(n+1)/(n+2) ;
%p end proc:
%t CoefficientList[ Series[1 + (HypergeometricPFQ[{1, 3/2, 5/2}, {3, 4}, 16 x] - 1), {x, 0, 17}], x]
%t Table[(2*n)!*(2*n+2)!/(2*n!*(n+1)!^2*(n+2)!),{n,30}] (* _Vincenzo Librandi_, Mar 25 2012 *)
%Y Cf. A000264, A000309.
%Y Equals A005568/2.
%Y Fourth row of array A102539.
%Y Column of array A073165.
%Y Image of A001700 under the "little Hankel" transform (see A056220 for definition). - _John W. Layman_, Aug 22 2000
%Y Cf. A000891.
%K easy,nonn,nice
%O 1,2
%A _N. J. A. Sloane_, _Simon Plouffe_
%E Better definition from _Michael Albert_, Oct 24 2008
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