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))))
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.
Original entry on oeis.org
812, 579667803, 235948437342837440, 52366358060537007928863206000, 6262970727027052056580468670430288168750000, 401820562589647140572840734882930708995214500792163500000000
Offset: 1
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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 *)
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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))))
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.