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A099117
Number of rhombus tilings of a hexagon with side lengths 2n+3,2n-1,2n+3,2n+3,2n-1,2n+3 which contain the rhombus above and next to the center of the hexagon.
5
152, 436381660, 574954797841668608, 388062759166540341977143692000, 137515819873369461005150742745259538637500000, 25797761881848486655895899589856317740988916476499759600000000
OFFSET
1,1
FORMULA
a(n) ~ exp(1/12) * 3^(137/12 + 30*n + 18*n^2) / (A * n^(1/12) * 2^(131/6 + 40*n + 24*n^2)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 29 2023
MATHEMATICA
a[n_] := (1/3+2(6n^2+9n+2)/(n+1)^2 Binomial[2n, n]^3/Binomial[6n+4, 3n+2]) Product[(i+j+k-1)/(i+j+k-2), {i, 1, 2n+3}, {j, 1, 2n-1}, {k, 1, 2n+3}];
Array[a, 6] (* Jean-François Alcover, Nov 18 2018, from PARI *)
PROG
(PARI) a(n)=(1/3+2*(6*n*n+9*n+2)/(n+1)^2*binomial(2*n, n)^3/binomial(6*n+4, 3*n+2))*prod(i=1, 2*n+3, prod(j=1, 2*n-1, prod(k=1, 2*n+3, (i+j+k-1)/(i+j+k-2))))
CROSSREFS
KEYWORD
nonn
AUTHOR
Ralf Stephan, Oct 01 2004
STATUS
approved