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%I #26 Jul 20 2020 02:04:59
%S 1,2,84,81796,1844536720,962310111888300,11608208114358751650000,
%T 3236574482779383546336417240000,
%U 20853456581643133066208521560263633137920,3104385823530881109001458753652585998600603921849920,10676554307318599842868990948461304923921623250562199975300214736
%N Number of transpose-complementary plane partitions of n.
%C The complement of a plane partition inside an m X m X m cube consists of the boxes which are within the cube, but not in the plane partition, rotated in an appropriate way.
%C a(n) is the number of plane partitions inside an 2n X 2n X 2n cube whose (matrix) transpose when written as an 2n X 2n array is the same as its complement.
%H R. P. Stanley, <a href="http://dx.doi.org/10.1016/0097-3165(86)90028-2">Symmetries of Plane Partitions</a>, J. Comb. Theory Ser. A 43 (1986), 103-113.
%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="http://en.wikipedia.org/wiki/Plane_partition">Plane partition</a>
%F a(n) = binomial(3n-1,n)*Product(i=1..2n-2,Product(j=i..2n-2,(2n+i+j+1)/(i+j+1))).
%F a(n) ~ exp(1/24) * 3^(9*n^2 - 3*n/2 - 1/24) / (sqrt(A) * n^(1/24) * 2^(12*n^2 - n - 1/3)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Feb 28 2015
%e When n=2, there are two transpose-complementary plane partitions,
%e [1 1] and [2 1], both of whose transpose and complement is equal to themselves.
%e [1 1] [1 0]
%t Table[Binomial[3n-1,n]Product[(2n+i+j+1)/(i+j+1),{i,1,2n-2}, {j,i,2n-2}], {n,0,10}] (* _Harvey P. Dale_, Jan 27 2012 *)
%o (PARI) a(n) = binomial(3*n-1,n)*prod(i=1,2*n-2,prod(j=i,2*n-2,(2*n+i+j+1)/(i+j+1))); \\ _Michel Marcus_, Jun 18 2015
%Y Cf. A008793, A051255, A078920, A123352.
%K nonn,nice
%O 0,2
%A _Arvind Ayyer_, Jan 21 2011
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