A099116 Number of rhombus tilings of a hexagon with side lengths 2n+2,2n,2n+2,2n+2,2n,2n+2 which contain the rhombus above and next to the center of the hexagon.

812, 579667803, 235948437342837440, 52366358060537007928863206000, 6262970727027052056580468670430288168750000, 401820562589647140572840734882930708995214500792163500000000

Offset: 1

Author

Ralf Stephan, Oct 01 2004

Keywords

• nonn

Links

• M. Fulmek and C. Krattenthaler, The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis, II, arXiv:math/9909038 [math.CO], 1999.

Crossrefs

Cf. A099112, A099113, A099114, A099115, A099117, A008793.

Programs

Mathematica

G = BarnesG; a[n_] := (G[2n+1]^(-2n-1) (G[2n+2] G[2n+4])^(2(n+1)) G[6n+5]( (4 Binomial[2n, n]^3)/Binomial[6n+4, 3n+2] + 1/3))/(G[2n+3]^(2n) (Gamma[ 2n+1] Gamma[2n+3])^(2(n+1))(G[4n+3]^2 G[4n+5])); Array[a, 6] (* Jean-François Alcover, Feb 20 2019 *)

PARI

a(n)=(1/3+4*binomial(2*n,n)^3/binomial(6*n+4,3*n+2))*prod(i=1,2*n+2,prod(j=1,2*n,prod(k=1,2*n+2,(i+j+k-1)/(i+j+k-2))))

Formula

a(n) ~ exp(1/12) * 3^(83/12 + 24*n + 18*n^2) / (A * n^(1/12) * 2^(71/6 + 32*n + 24*n^2)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 29 2023