A049504 a(n) = Product_{i = 0..n-1} ((3*i+1)!*(6*i)!*(2*i)!)/((4*i)!*(4*i+1)!).

1, 1, 12, 47520, 266499072000, 5578457158440714240000, 903833169262981594760400076800000000, 2035652583056655211566004660439314466655436800000000000, 103962610930356904475854868257296244089884364267142052118842572800000000000000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

Given in first printing of Bressoud book as number of cyclically symmetric transpose complement plane partitions. For correct version see A051255.

References

• D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.15), p. 199.

Programs

Maple

a := proc(n) local i; mul((3*i+1)!*(6*i)!*(2*i)!/((4*i)!*(4*i+1)!),i = 0..n-1); end;

Mathematica

Table[Product[((3i+1)!(6i)!(2i)!)/((4i)!(4i+1)!),{i,0,n-1}],{n,0,10}] (* Harvey P. Dale, Apr 25 2016 *)

Formula

a(n) ~ A^(-1/2) * Gamma(1/3) * 2^(-1/9 + 3*n/2 - 4*n^2) * 3^(-1/24 - 5*n/2 + 9*n^2/2) * exp(1/24 + n - 9*n^2/4) * n^(1/8 - n + 3*n^2/2) * Pi^((n-1)/2), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Apr 25 2016

Extensions

Definition corrected by Harvey P. Dale, Apr 25 2016