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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 (list; graph; refs; listen; history; text; internal format)
 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 T. D. Noe, Table of n, a(n) for n = 0..40 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. Cf. A005157, A008793. Sequence in context: A062499 A305854 A234296 * A136518 A168369 A363206 Adjacent sequences: A049502 A049503 A049504 * A049506 A049507 A049508 KEYWORD nonn,easy AUTHOR N. J. A. Sloane EXTENSIONS Edited by N. J. A. Sloane, Jun 30 2013; codes and formula checked by N. J. A. Sloane and Olivier Gérard STATUS approved

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Last modified September 22 09:12 EDT 2023. Contains 365520 sequences. (Running on oeis4.)