A099115 Number of rhombus tilings of a hexagon with side lengths 2n+1,2n-1,2n+1,2n+1,2n-1,2n+1 which contain the rhombus above and next to the center of the hexagon.

11, 325908, 5604277805984, 53038629767258343852608, 271847253225677708645983929633862500, 749641889501430920151272774045675453348280000000000

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^(-7/12 + 6*n + 18*n^2) / (A * n^(1/12) * 2^(11/6 + 8*n + 24*n^2)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 29 2023

Mathematica

G = BarnesG; a[n_] := (G[2n+2]^(1-2n) (G[2n+1] G[2n+3])^(2n+1) G[6n+2] ((( 10n+3) Binomial[2n, n]^3)/(n Binomial[6n, 3n]) + 8) Gamma[2n+2]^(-2n-1))/((G[2n] Gamma[2n])^(2n) (24 G[4n+1]^2 G[4n+3] Gamma[2n]));
Array[a, 6] (* Jean-François Alcover, Feb 20 2019 *)

Prog

(PARI) a(n)=(1/3+(10*n+3)/(24*n)*binomial(2*n, n)^3/binomial(6*n, 3*n))*prod(i=1, 2*n+1, prod(j=1, 2*n-1, prod(k=1, 2*n+1, (i+j+k-1)/(i+j+k-2))))

Crossrefs

Cf. A099112, A099113, A099114, A099116, A099117, A008793.

Sequence in context: A098880 A076173 A100108 * A076171 A297057 A090102

Adjacent sequences: A099112 A099113 A099114 * A099116 A099117 A099118

Keyword

nonn

Author

Ralf Stephan, Oct 01 2004

Status

approved