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

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

Michael De Vlieger, Table of n, a(n) for n = 1..1659

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, Publications and Preprints.

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.

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)

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

Crossrefs

Cf. A000984, A001622, A001700, A002457.

Cf. A092437, A092438, A092439, A092440, A092441, A092442.

Sequence in context: A037765 A037653 A229665 * A356280 A108080 A113441

Adjacent sequences: A092440 A092441 A092442 * A092444 A092445 A092446

Keyword

easy,nonn

Author

Christopher Hanusa (chanusa(AT)math.washington.edu), Mar 24 2004

Status

approved