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A008793
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Number of ways to tile hexagon of edge n with diamonds of side 1. Also number of plane partitions whose Young diagrams fit inside an n X n X n box.
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21
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1, 2, 20, 980, 232848, 267227532, 1478619421136, 39405996318420160, 5055160684040254910720, 3120344782196754906063540800, 9265037718181937012241727284450000, 132307448895406086706107959899799334375000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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REFERENCES
| Gordon G. Cash and Jerry Ray Dias, Computation, Properties and Resonance Topology of Benzenoid Monoradicals and Polyradicals and the Eigenvectors Belonging to Their Zero Eigenvalues, J. Math. Chem., 30 (2001), 429-444. [See K, p. 442.]
J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see p. 261).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..30
T. Amdeberhan, V. H. Moll, Arithmetic properties of plane partitions, El. J. Comb. 18 (2) (2011) # P1
P. Di Francesco, P. Zinn-Justin and J.-B. Zuber, Determinant formulae for some tiling problems...
I. Fischer, [math/9906102] Enumeration of rhombus tilings of a hexagon which contain a fixed rhombus in the center
P. J. Forrester and A. Gamburd, Counting formulas associated with some random matrix averages
M. Fulmek and C. Krattenthaler, [math/9909038] The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis, II
H. Helfgott and I. M. Gessel, Enumeration of tilings of diamonds and hexagons with defects
C. Krattenthaler, Advanced Determinant Calculus: A Complement, Linear Algebra Appl. 411 (2005), 68-166; arXiv:math/0503507v2 [math.CO].
P. A. MacMahon, Combinatory Analysis, vol. 2, Cambridge University Press, 1916; reprinted by Chelsea, New York, 1960.
J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics
J. Propp, Updated article
N. C. Saldanha and C. Tomei, [math/9801111] An overview of domino and lozenge tilings
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
| Product_{i=0..n-1} (i^(-i)*(n+i)^(2i-n)*(2n+i)^(n-i)).
Product_{i=1..n} Product_{j=0..n-1} (3*n-i-j)/(2*n-i-j).
Product[Gamma[i]Gamma[i+2n]/Gamma[i+n]^2, {i, n}
Product[i=0..n-1, i!(i+2n)!/(i+n)!^2 ].
a(n)=Prod[i=1..n, Prod[j=n..2n-1, i+j]/Prod[j=0..n-1, i+j]]; - Paul Barry (pbarry(AT)wit.ie), Jun 13 2006
For n >= 1, a(n) = det(binomial(2*n,n+i-j))1<=i,j<=n [Krattenhaller, Theorem 4, with a = b = c = n].
Let H(n) = product {k = 1..n-1} k!. Then for a,b,c nonnegative integers (H(a)*H(b)*H(c)*H(a+b+c))/(H(a+b)*H(b+c)*H(c+a)) is an integer [MacMahon, Section 4.29 with x -> 1]. Setting a = b = c = n gives the entries for this sequence. - Peter Bala, Dec 22 2011
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MAPLE
| A008793 := proc(n) local i; mul((i - 1)!*(i + 2*n - 1)!/((i + n - 1)!)^2, i = 1 .. n) end proc;
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MATHEMATICA
| Table[ Product[ (i+j+k-1)/(i+j+k-2), {i, n}, {j, n}, {k, n} ], {n, 10} ]
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CROSSREFS
| Cf. A066931. Main diagonal of array A103905.
Sequence in context: A006547 A135757 A158843 * A015192 A012790 A013144
Adjacent sequences: A008790 A008791 A008792 * A008794 A008795 A008796
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Jonas Wallgren (jwc(AT)ida.liu.se)
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EXTENSIONS
| More terms from Eric Weisstein (eric(AT)weisstein.com)
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