A049503 - OEIS

%I #40 Sep 08 2022 08:44:58

%S 1,1,4,49,1764,184041,55294096,47675849104,117727187246656,

%T 831443906113411600,16779127803917965290000,

%U 966945347924006310543140625,159045186822042363450404006250000,74638947576233124529271587010756250000,99910846988474589225795290311922220324000000

%N a(n) = A005130(n)^2.

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

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

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

%H Seiichi Manyama, <a href="/A049503/b049503.txt">Table of n, a(n) for n = 0..66</a>

%H M. Ciucu, <a href="http://dx.doi.org/10.1006/jcta.1998.2922">The equivalence between enumerating cyclically symmetric, self-complementary and totally symmetric, self-complementary plane partitions</a>, J. Combin. Theory Ser. A 86 (1999), 382-389.

%H G. Kuperberg, <a href="http://dx.doi.org/10.1016/0097-3165(94)90094-9">Symmetries of plane partitions and the permanent-determinant method</a>, J. Comb. Theory Ser. A, 68 (1994), 115-151. [From _Peter J. Taylor_, Jun 17 2015]

%H G. Kuperberg, <a href="http://arxiv.org/abs/math/0008184">Symmetry classes of alternating-sign matrices under one roof</a>, arXiv:math/0008184 [math.CO], 2000-2001. [Th. 5].

%H P. J. Taylor, <a href="http://cheddarmonk.org/papers/distinct-dimer-hex-tilings.pdf">Counting distinct dimer hex tilings</a>, Preprint, 2015.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Plane_partition">Plane partition</a>

%F 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]

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

%o (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

%o (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

%o (Python)

%o from math import prod, factorial

%o 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

%Y Cf. A005130.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_