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A049505
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a(n) = Product_{1<=i<=j<=n} (i+j+n-1)/(i+j-1), number of symmetric plane partitions in n-cube.
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6
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1, 2, 10, 112, 2772, 151008, 18076916, 4751252480, 2740612658576, 3468301123758080, 9627912669442441500, 58618653300361405440000, 782683432110638830001250000, 22916694891747599820616089600000, 1471328419282772010324439370939640000
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OFFSET
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0,2
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COMMENTS
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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.
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REFERENCES
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D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.8), p. 198.
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LINKS
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FORMULA
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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
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)
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MAPLE
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A049505 := proc(n) local i, j; mul(mul((i+j+n-1)/(i+j-1), j=i..n), i=1..n); end;
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n)=prod(i=1, n, prod(j=i, n, (i+j+n-1)/(i+j-1)))
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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