A071096 Number of ways to tile hexagon of edges n, n+1, n+2, n, n+1, n+2 with diamonds of side 1.

1, 10, 490, 116424, 133613766, 739309710568, 19702998159210080, 2527580342020127455360, 1560172391098377453031770400, 4632518859090968506120863642225000, 66153724447703043353053979949899667187500, 4542776083800437392420665771479758969781250000000, 1499928882906010042230116408158354282455601808812500000000

Offset

0,2

References

J. 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 page 261).

Links

Table of n, a(n) for n=0..12.

J. Propp, Updated article

J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics

Formula

Product_{i=0..a-1} Product_{j=0..b-1} Product_{k=0..c-1} (i+j+k+2)/(i+j+k+1) with a=n, b=n+1, c=n+2.

a(n) = (-1)^floor((n+1)/2)*det(M(n+1)) where M(n) is the n X n matrix m(i, j)=C(2n, i+j), with i and j ranging from 1 to n. - Benoit Cloitre, Jan 31 2003

a(n) = (1/2)*Product[Product[Product[(i+j+k-1)/(i+j+k-2),{i,1,n+1}],{j,1,n+1}],{k,1,n+1}]. a(n) = A008793(n+1)/2. - Alexander Adamchuk, Jul 10 2006

a(n) ~ exp(1/12) * 3^(9*n^2/2 + 9*n + 53/12) / (A * n^(1/12) * 2^(6*n^2 + 12*n + 27/4)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Apr 26 2015

Mathematica

Table[Product[Product[Product[(i+j+k-1)/(i+j+k-2), {i, 1, n+1}], {j, 1, n+1}], {k, 1, n+1}], {n, 0, 10}]/2 (* Alexander Adamchuk, Jul 10 2006 *)

Prog

(PARI) {a(n) = abs(matdet(matrix(n+1, n+1, i, j, binomial(2*(n+1), i+j))))}; \\ Shifted by Georg Fischer, Jun 19 2022

Crossrefs

Cf. A008793, A071094, A071095.

Sequence in context: A035320 A200458 A223122 * A039835 A293090 A127947

Adjacent sequences: A071093 A071094 A071095 * A071097 A071098 A071099

Keyword

nonn

Author

N. J. A. Sloane, May 28 2002

Status

approved