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A092443 Sequence arising from enumeration of domino tilings of Aztec Pillow-like regions.

%I A092443 #48 Sep 10 2024 12:38:10
%S A092443 3,12,50,210,882,3696,15444,64350,267410,1108536,4585308,18929092,
%T A092443 78004500,320932800,1318498920,5409723510,22169259090,90751353000,
%U A092443 371125269900,1516311817020,6189965556060,25249187564640,102917884095000,419218847880300,1706543186909652
%N A092443 Sequence arising from enumeration of domino tilings of Aztec Pillow-like regions.
%C A092443 The sequence 1, 3, 12, 50, ... is ((n+2)/2)*C(2n,n) with g.f. F(1/2,3;2;4x). - _Paul Barry_, Sep 18 2008
%D A092443 James Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 13).
%H A092443 Michael De Vlieger, <a href="/A092443/b092443.txt">Table of n, a(n) for n = 1..1659</a>
%H A092443 Sanjay Moudgalya, Abhinav Prem, Rahul Nandkishore, Nicolas Regnault, and B. Andrei Bernevig, <a href="https://arxiv.org/abs/1910.14048">Thermalization and its absence within Krylov subspaces of a constrained Hamiltonian</a>, arXiv:1910.14048 [cond-mat.str-el], 2019.
%H A092443 James Propp, <a href="http://faculty.uml.edu/jpropp/articles.html">Publications and Preprints</a>.
%H A092443 James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), <a href="https://library.slmath.org/books/Book38/index.html">New Perspectives in Algebraic Combinatorics</a>, Cambridge University Press, Cambridge, 1999, pp. 255-291.
%H A092443 Eric Rowland and Jason Wu, <a href="https://arxiv.org/abs/2409.02789">The entries of the Sinkhorn limit of an m X n matrix</a>, arXiv:2409.02789 [math.NT], 2024. See p. 11.
%F A092443 a(n) = (2*n-1)!/((n-1)!)^2+(2*n)!/(n!)^2 = A002457(n-1) + A000984(n).
%F A092443 a(n) = (n+2)*A001700(n-1). - _Vladeta Jovovic_, Jul 12 2004
%F A092443 n*a(n) + (-7*n+4)*a(n-1) + 6*(2*n-3)*a(n-2) = 0. - _R. J. Mathar_, Nov 30 2012
%F A092443 From _Amiram Eldar_, Jan 27 2024: (Start)
%F A092443 Sum_{n>=1} 1/a(n) = 4*Pi*(11*sqrt(3)-3*Pi)/9 - 13.
%F A092443 Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(phi)*(13*sqrt(5)-30*log(phi))/5 - 11, where phi is the golden ratio (A001622). (End)
%F A092443 From _Peter Bala_, Aug 02 2024: (Start)
%F A092443 a(n) = 1/(n + 1)^2 * Sum_{k = 1..n+1} (k^3)*binomial(n+1, k)^2 = hypergeom([2, -n, -n], [1, 1], 1).
%F A092443 a(n) = 2*(n + 2)*(2*n - 1)/(n*(n + 1)) * a(n-1) with a(1) = 3. (End)
%e A092443 a(3) = 5!/2!2! + 6!/3!3! = 50.
%t A092443 Array[Binomial[2 # + 1, # + 1] &[# - 1]*(# + 2) &, 22] (* _Michael De Vlieger_, Dec 17 2017 *)
%o A092443 (MuPAD) combinat::catalan(n) *binomial(n+2,2) $ n = 1..22 // _Zerinvary Lajos_, Feb 15 2007
%o A092443 (PARI) a(n) = (n+2)*binomial(2*n-1, n); \\ _Altug Alkan_, Dec 17 2017
%Y A092443 Cf. A000984, A001622, A001700, A002457.
%Y A092443 Cf. A092437, A092438, A092439, A092440, A092441, A092442.
%K A092443 easy,nonn
%O A092443 1,1
%A A092443 Christopher Hanusa (chanusa(AT)math.washington.edu), Mar 24 2004