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A092443
Sequence arising from enumeration of domino tilings of Aztec Pillow-like regions.
8
3, 12, 50, 210, 882, 3696, 15444, 64350, 267410, 1108536, 4585308, 18929092, 78004500, 320932800, 1318498920, 5409723510, 22169259090, 90751353000, 371125269900, 1516311817020, 6189965556060, 25249187564640, 102917884095000, 419218847880300, 1706543186909652
OFFSET
1,1
COMMENTS
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
REFERENCES
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).
LINKS
Sanjay Moudgalya, Abhinav Prem, Rahul Nandkishore, Nicolas Regnault, and B. Andrei Bernevig, Thermalization and its absence within Krylov subspaces of a constrained Hamiltonian, arXiv:1910.14048 [cond-mat.str-el], 2019.
James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics, Cambridge University Press, Cambridge, 1999, pp. 255-291.
Eric Rowland and Jason Wu, The entries of the Sinkhorn limit of an m X n matrix, arXiv:2409.02789 [math.NT], 2024. See p. 11.
FORMULA
a(n) = (2*n-1)!/((n-1)!)^2+(2*n)!/(n!)^2 = A002457(n-1) + A000984(n).
a(n) = (n+2)*A001700(n-1). - Vladeta Jovovic, Jul 12 2004
n*a(n) + (-7*n+4)*a(n-1) + 6*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 30 2012
From Amiram Eldar, Jan 27 2024: (Start)
Sum_{n>=1} 1/a(n) = 4*Pi*(11*sqrt(3)-3*Pi)/9 - 13.
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)
From Peter Bala, Aug 02 2024: (Start)
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).
a(n) = 2*(n + 2)*(2*n - 1)/(n*(n + 1)) * a(n-1) with a(1) = 3. (End)
EXAMPLE
a(3) = 5!/2!2! + 6!/3!3! = 50.
MATHEMATICA
Array[Binomial[2 # + 1, # + 1] &[# - 1]*(# + 2) &, 22] (* Michael De Vlieger, Dec 17 2017 *)
PROG
(MuPAD) combinat::catalan(n) *binomial(n+2, 2) $ n = 1..22 // Zerinvary Lajos, Feb 15 2007
(PARI) a(n) = (n+2)*binomial(2*n-1, n); \\ Altug Alkan, Dec 17 2017
KEYWORD
easy,nonn
AUTHOR
Christopher Hanusa (chanusa(AT)math.washington.edu), Mar 24 2004
STATUS
approved