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A049505
a(n) = Product_{1<=i<=j<=n} (i+j+n-1)/(i+j-1), number of symmetric plane partitions in n-cube.
6
1, 2, 10, 112, 2772, 151008, 18076916, 4751252480, 2740612658576, 3468301123758080, 9627912669442441500, 58618653300361405440000, 782683432110638830001250000, 22916694891747599820616089600000, 1471328419282772010324439370939640000
OFFSET
0,2
COMMENTS
The first printing of the Bressoud book states that the formula Product_{1<=i<=j<=n} (i+j+n-1)/(i+j-1) in Eq. (6.8) is the number of totally symmetric plane partitions. This is wrong, although it does produce the current sequence. For the correct formula for the number of totally symmetric plane partitions see A005157.
REFERENCES
D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.8), p. 198.
LINKS
P. J. Taylor, Counting distinct dimer hex tilings, Preprint, 2015.
P. J. Taylor, Counting distinct dimer hex tilings, arXiv:1602.06796 [math.CO], 2016.
FORMULA
a(n) = Product_{1<=i<=j<=n} (i+j+n-1)/(i+j-1).
a(n) = Product_{i=1..n} (((2*i-2)!*(i+2*n-1)!)/((i+n-1)!*(2*i+n-2)!)). - Jean-François Alcover, Jun 22 2012
a(n) = Product_{i=1..n} (binomial((i-1) + 2*n, n)/binomial(n + 2*(i-1), n)). - Olivier Gérard, Feb 25 2015
a(n) ~ exp(1/24) * 3^(9*n^2/4 + 3*n/4 - 1/24) / (A^(1/2) * n^(1/24) * 2^(3*n^2 + n/2 + 1/8)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 01 2015
From Peter Bala, Feb 15 2023: (Start)
a(n+1) = m(n)*a(n) where m(n) = ((3*n + 2)!*n!^2)/((2 n)!*(2 n + 1)!^2) * Product_{i = 1..n} n + 2*i for n >= 1.
Conjectures:
1) the supercongruence a(p) == 2^((p+1)/2) (mod p^3) holds for all primes p >= 3 (checked up to p = 1009).
2) the congruence a(p^2) == (-1)^((p^2-1)/8)*a(p)^(p^2-p+1) (mod p^3) holds for all primes p >= 3 (checked up to p = 89).
3) the congruence a(p^3) == a(p^2)^((p^3-p^2+2)/2) (mod p^3) holds for all primes p >= 2. (End)
MAPLE
A049505 := proc(n) local i, j; mul(mul((i+j+n-1)/(i+j-1), j=i..n), i=1..n); end;
MATHEMATICA
a[n_] := Product[ ((2i-2)!*(i+2n-1)!)/((i+n-1)!*(2i+n-2)!), {i, 1, n}]; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Jun 22 2012, after PARI *)
PROG
(PARI) a(n)=prod(i=1, n, prod(j=i, n, (i+j+n-1)/(i+j-1)))
CROSSREFS
Main diagonal of array A102539.
Main diagonal of array in A073165.
Sequence in context: A062499 A305854 A234296 * A136518 A168369 A363206
KEYWORD
nonn,easy
EXTENSIONS
Edited by N. J. A. Sloane, Jun 30 2013; codes and formula checked by N. J. A. Sloane and Olivier Gérard
STATUS
approved