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

1,1

Links

Table of n, a(n) for n=1..6.

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.

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

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 *)

Prog

(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))))

Crossrefs

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

Sequence in context: A252223 A158667 A035854 * A183820 A272606 A235976

Adjacent sequences: A099113 A099114 A099115 * A099117 A099118 A099119

Keyword

nonn

Author

Ralf Stephan, Oct 01 2004

Status

approved