A049503 a(n) = A005130(n)^2.

1, 1, 4, 49, 1764, 184041, 55294096, 47675849104, 117727187246656, 831443906113411600, 16779127803917965290000, 966945347924006310543140625, 159045186822042363450404006250000, 74638947576233124529271587010756250000, 99910846988474589225795290311922220324000000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Expansion of generating function A_{QT}^(1)(4n).

a(n) is the number of cyclically symmetric and self-complementary plane partitions in a (2n)-cube. - Peter J. Taylor, Jun 17 2015

References

• D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.16), p. 199.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..66
• M. Ciucu, The equivalence between enumerating cyclically symmetric, self-complementary and totally symmetric, self-complementary plane partitions, J. Combin. Theory Ser. A 86 (1999), 382-389.
• G. Kuperberg, Symmetries of plane partitions and the permanent-determinant method, J. Comb. Theory Ser. A, 68 (1994), 115-151. [From _Peter J. Taylor_, Jun 17 2015]
• G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2000-2001. [Th. 5].
• P. J. Taylor, Counting distinct dimer hex tilings, Preprint, 2015.
• Wikipedia, Plane partition

Crossrefs

Cf. A005130.

Programs

Magma

[n eq 0 select 1 else &*[(Factorial(3*k+1)/Factorial(n+k))^2: k in [0..n-1]]: n in [0..15]]; // Bruno Berselli, Jun 23 2015

Mathematica

f[n_]:=Product[((3 k + 1)!/(n + k)!)^2, {k, 0, n-1}]; Table[f[n], {n, 0, 15}] (* Vincenzo Librandi, Jun 18 2015 *)

PARI

a(n) = 2^n*matdet(matrix(n, n, i, j, i--; j--; binomial(i+j, 2*i-j)/2+binomial(i+j, 2*i-j-1))); \\ Michel Marcus, Jun 18 2015

Python

from math import prod, factorial
def A049503(n): return (prod(factorial(3*k+1) for k in range(n))//prod(factorial(n+k) for k in range(n)))**2 # Chai Wah Wu, Feb 02 2022

Formula

a(n) = 2^n * det U(n), where U(n) is the n X n matrix with entry (i, j) equal to binomial(i+j, 2*i-j)/2 + binomial(i+j, 2*i-j-1). [Ciucu]