cp's OEIS Frontend

A071094 Number of ways to tile hexagon of edges n, n, n+1, n, n, n+1 with diamonds of side 1.

%I A071094 #28 Apr 03 2021 08:37:00
%S A071094 1,3,50,4116,1646568,3184461423,29706808370096,1335119245893326400,
%T A071094 288882990167192721013376,300792059519113653077154558000,
%U A071094 1506680146887473588202049621593937500,36298820709557430183399305000196605531250000,4205446372314569673006362329181090368935937500000000,2342761095072644391194625697884219372917666852341417500000000
%N A071094 Number of ways to tile hexagon of edges n, n, n+1, n, n, n+1 with diamonds of side 1.
%D A071094 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).
%H A071094 Seiichi Manyama, <a href="/A071094/b071094.txt">Table of n, a(n) for n = 0..53</a>
%H A071094 J. Propp, <a href="http://faculty.uml.edu/jpropp/update.pdf">Updated article</a>
%H A071094 J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), <a href="http://www.msri.org/publications/books/Book38/contents.html">New Perspectives in Algebraic Combinatorics</a>
%F A071094 a(n) = 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=b=n, c=n+1.
%F A071094 a(n) = Product_{k=0..n} C(2n+k,n+k)/C(n+k,k). - _Paul Barry_, May 13 2008
%F A071094 a(n) ~ exp(1/12) * 3^(9*n^2/2 + 3*n + 5/12) / (A * n^(1/12) * 2^(6*n^2 + 4*n + 3/4)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Apr 26 2015
%t A071094 Table[Product[(i+j+k+2)/(i+j+k+1),{i,0,n-1},{j,0,n-1},{k,0,n}],{n,0,15}] (* _Vaclav Kotesovec_, Apr 26 2015 *)
%o A071094 (PARI) a(n) = prod(k=0, n, binomial(2*n+k,n+k)/binomial(n+k,k)) \\ _Michel Marcus_, May 20 2013
%Y A071094 Cf. A008793, A071095, A103905.
%K A071094 nonn,easy
%O A071094 0,2
%A A071094 _N. J. A. Sloane_, May 28 2002