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%I #41 Feb 19 2023 09:18:35
%S 1,2,10,112,2772,151008,18076916,4751252480,2740612658576,
%T 3468301123758080,9627912669442441500,58618653300361405440000,
%U 782683432110638830001250000,22916694891747599820616089600000,1471328419282772010324439370939640000
%N a(n) = Product_{1<=i<=j<=n} (i+j+n-1)/(i+j-1), number of symmetric plane partitions in n-cube.
%C 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.
%D D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.8), p. 198.
%H T. D. Noe, <a href="/A049505/b049505.txt">Table of n, a(n) for n = 0..40</a>
%H P. J. Taylor, <a href="http://cheddarmonk.org/papers/distinct-dimer-hex-tilings.pdf">Counting distinct dimer hex tilings</a>, Preprint, 2015.
%H P. J. Taylor, <a href="https://arxiv.org/abs/1602.06796">Counting distinct dimer hex tilings</a>, arXiv:1602.06796 [math.CO], 2016.
%F a(n) = Product_{1<=i<=j<=n} (i+j+n-1)/(i+j-1).
%F 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
%F a(n) = Product_{i=1..n} (binomial((i-1) + 2*n, n)/binomial(n + 2*(i-1), n)). - _Olivier Gérard_, Feb 25 2015
%F 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
%F From _Peter Bala_, Feb 15 2023: (Start)
%F 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.
%F Conjectures:
%F 1) the supercongruence a(p) == 2^((p+1)/2) (mod p^3) holds for all primes p >= 3 (checked up to p = 1009).
%F 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).
%F 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)
%p A049505 := proc(n) local i,j; mul(mul((i+j+n-1)/(i+j-1),j=i..n),i=1..n); end;
%t 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 *)
%o (PARI) a(n)=prod(i=1,n,prod(j=i,n,(i+j+n-1)/(i+j-1)))
%Y Main diagonal of array A102539.
%Y Main diagonal of array in A073165.
%Y Cf. A005157, A008793.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E Edited by _N. J. A. Sloane_, Jun 30 2013; codes and formula checked by _N. J. A. Sloane_ and _Olivier Gérard_
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