A099112
Number of rhombus tilings of a hexagon with all sides of length 2n which contain the rhombus above and next to the center of the hexagon.
Original entry on oeis.org
6, 73080, 472598638512, 1631619756904447290240, 3008692066440440678503082183460000, 2962701176869736970134706082584757742017500000000, 1557551298812773746701490125169378658941648550102913633903040000000
Offset: 1
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a[n_] := (1/3 - 1/12 Binomial[2n, n]^3/Binomial[6n, 3n]) Product[(i + j + k - 1)/(i + j + k - 2), {i, 1, 2n}, {j, 1, 2n}, {k, 1, 2n}];
Array[a, 6] (* Jean-François Alcover, Nov 18 2018 *)
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a(n)=(1/3-1/12*binomial(2*n,n)^3/binomial(6*n,3*n))*prod(i=1,2*n,prod(j=1,2*n,prod(k=1,2*n,(i+j+k-1)/(i+j+k-2))))
A099113
Number of rhombus tilings of a hexagon with all sides of length 2n+1 which contain the rhombus above and next to the center of the hexagon.
Original entry on oeis.org
364, 94682016, 13704096621766720, 1074416738842280125146121600, 45276656003305722314718295417920118125000, 1022271041965503132822786100650613600920143229195000000000
Offset: 1
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G = BarnesG; a[n_] := (G[2n+2]^(1-4n) G[2n+3]^(4n+2) G[6n+4] (Binomial[2n, n]^3/Binomial[6n+2, 3n+1]+1) Gamma[2n+2]^(-4n-2))/(3G[4n+3]^3); Array[a, 6] (* Jean-François Alcover, Feb 20 2019 *)
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a(n)=(1/3+1/3*binomial(2*n,n)^3/binomial(6*n+2,3*n+1))*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))))
A099114
Number of rhombus tilings of a hexagon with side lengths 2n,2n+2,2n,2n,2n+2,2n which contain the rhombus above and next to the center of the hexagon.
Original entry on oeis.org
19, 2252052, 125575149020464, 3624877924928635307525440, 55204136490632334691332479792745796875, 446207680704793917097310140821019734826847707500000000
Offset: 1
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G = BarnesG; a[n_] := (G[2n+1]^(2-2n) G[2n+3]^(1-2n)(G[2n+2] G[2n+4])^(2n) G[6n+3](1/3 - ((10n+2) Binomial[2n, n]^3)/((6n+3) Binomial[6n+2, 3n+1]))) /((G[4n+1] G[4n+3]^2) (Gamma[2n+1] Gamma[2n+3])^(2n)); Array[a, 6] (* Jean-François Alcover, Feb 20 2019 *)
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a(n)=(1/3-(10*n+2)/(6*n+3)*binomial(2*n,n)^3/binomial(6*n+2,3*n+1))*prod(i=1,2*n,prod(j=1,2*n+2,prod(k=1,2*n,(i+j+k-1)/(i+j+k-2))))
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.
Original entry on oeis.org
11, 325908, 5604277805984, 53038629767258343852608, 271847253225677708645983929633862500, 749641889501430920151272774045675453348280000000000
Offset: 1
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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 *)
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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))))
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.
Original entry on oeis.org
152, 436381660, 574954797841668608, 388062759166540341977143692000, 137515819873369461005150742745259538637500000, 25797761881848486655895899589856317740988916476499759600000000
Offset: 1
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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 *)
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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))))
Showing 1-5 of 5 results.