cp's OEIS Frontend

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.

%I A099115 #11 Aug 29 2023 12:24:01
%S A099115 11,325908,5604277805984,53038629767258343852608,
%T A099115 271847253225677708645983929633862500,
%U A099115 749641889501430920151272774045675453348280000000000
%N 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.
%H A099115 M. Fulmek and C. Krattenthaler, <a href="https://arxiv.org/abs/math/9909038">The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis, II</a>, arXiv:math/9909038 [math.CO], 1999.
%F A099115 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
%t A099115 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]));
%t A099115 Array[a, 6] (* _Jean-François Alcover_, Feb 20 2019 *)
%o A099115 (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))))
%Y A099115 Cf. A099112, A099113, A099114, A099116, A099117, A008793.
%K A099115 nonn
%O A099115 1,1
%A A099115 _Ralf Stephan_, Oct 01 2004