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A008793 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. 27
1, 2, 20, 980, 232848, 267227532, 1478619421136, 39405996318420160, 5055160684040254910720, 3120344782196754906063540800, 9265037718181937012241727284450000, 132307448895406086706107959899799334375000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The 3-dimensional analog of A000984. - William Entriken, Aug 06 2013

REFERENCES

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.8), p. 198. The first printing of Eq. (6.8) is wrong (see A049505 and A005157), but if one changes the limits in the formula (before it is corrected) to {1 <= i <= r, 1 <= j <= r}, one obtains the present sequence. - N. J. A. Sloane, Jun 30 2013

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).

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..54 (terms 0..30 from T. D. Noe)

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 and Application to Fully Packed Loops, arXiv:math-ph/0410002, 2004.

I. Fischer, Enumeration of rhombus tilings of a hexagon which contain a fixed rhombus in the center, arXiv:math/9906102 [math.CO], 1999.

P. J. Forrester and A. Gamburd, Counting formulas associated with some random matrix averages, arXiv:math/0503002 [math.CO], 2005.

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, arXiv:math/9909038 [math.CO], 1999.

I. Gutman, S. J. Cyvin, and V. Ivanov-Petrovic, Topological properties of circumcoronenes, Z. Naturforsch., 53a, 1998, 699-703 (see p. 700) - Emeric Deutsch, May 14 2018

H. Helfgott and I. M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.

C. Krattenthaler, Advanced Determinant Calculus: A Complement, Linear Algebra Appl. 411 (2005), 68-166; arXiv:math/0503507v2 [math.CO], 2005.

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, An overview of domino and lozenge tilings, arXiv:math/9801111 [math.CO], 1998.

P. J. Taylor, Counting distinct dimer hex tilings, Preprint, 2015.

Eric Weisstein's World of Mathematics, Plane Partition.

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, 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

a(n) ~ exp(1/12) * 3^(9*n^2/2 - 1/12) / (A * n^(1/12) * 2^(6*n^2 - 1/4)), where A = A074962 = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Feb 27 2015

MAPLE

A008793 := proc(n) local i; mul((i - 1)!*(i + 2*n - 1)!/((i + n - 1)!)^2, i = 1 .. n) end proc;

MATHEMATICA

Table[ Product[ (i+j+k-1)/(i+j+k-2), {i, n}, {j, n}, {k, n} ], {n, 10} ]

CROSSREFS

Cf. A066931. Main diagonal of array A103905.

Sequence in context: A135757 A301945 A158843 * A015192 A012790 A273194

Adjacent sequences:  A008790 A008791 A008792 * A008794 A008795 A008796

KEYWORD

nonn,easy,nice

AUTHOR

Jonas Wallgren

EXTENSIONS

More terms from Eric W. Weisstein

STATUS

approved

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Last modified August 16 06:22 EDT 2018. Contains 313782 sequences. (Running on oeis4.)