This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A049505 #41 Feb 19 2023 09:18:35
%S A049505 1,2,10,112,2772,151008,18076916,4751252480,2740612658576,
%T A049505 3468301123758080,9627912669442441500,58618653300361405440000,
%U A049505 782683432110638830001250000,22916694891747599820616089600000,1471328419282772010324439370939640000
%N A049505 a(n) = Product_{1<=i<=j<=n} (i+j+n-1)/(i+j-1), number of symmetric plane partitions in n-cube.
%C A049505 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 A049505 D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.8), p. 198.
%H A049505 T. D. Noe, <a href="/A049505/b049505.txt">Table of n, a(n) for n = 0..40</a>
%H A049505 P. J. Taylor, <a href="http://cheddarmonk.org/papers/distinct-dimer-hex-tilings.pdf">Counting distinct dimer hex tilings</a>, Preprint, 2015.
%H A049505 P. J. Taylor, <a href="https://arxiv.org/abs/1602.06796">Counting distinct dimer hex tilings</a>, arXiv:1602.06796 [math.CO], 2016.
%F A049505 a(n) = Product_{1<=i<=j<=n} (i+j+n-1)/(i+j-1).
%F A049505 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 A049505 a(n) = Product_{i=1..n} (binomial((i-1) + 2*n, n)/binomial(n + 2*(i-1), n)). - _Olivier Gérard_, Feb 25 2015
%F A049505 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 A049505 From _Peter Bala_, Feb 15 2023: (Start)
%F A049505 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 A049505 Conjectures:
%F A049505 1) the supercongruence a(p) == 2^((p+1)/2) (mod p^3) holds for all primes p >= 3 (checked up to p = 1009).
%F A049505 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 A049505 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 A049505 := proc(n) local i,j; mul(mul((i+j+n-1)/(i+j-1),j=i..n),i=1..n); end;
%t A049505 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 A049505 (PARI) a(n)=prod(i=1,n,prod(j=i,n,(i+j+n-1)/(i+j-1)))
%Y A049505 Main diagonal of array A102539.
%Y A049505 Main diagonal of array in A073165.
%Y A049505 Cf. A005157, A008793.
%K A049505 nonn,easy
%O A049505 0,2
%A A049505 _N. J. A. Sloane_
%E A049505 Edited by _N. J. A. Sloane_, Jun 30 2013; codes and formula checked by _N. J. A. Sloane_ and _Olivier Gérard_