|
%I #23 Jun 02 2022 03:26:12
%S 0,1,0,2,4,1,9,80,82,2,44,4752,43390,4740,13,265,440192,59216968,
%T 59216648,439794,80,1854,59245120,164806652728,2649391488016,
%U 164806435822,59216644,579,14833,10930514688,817056761525488,312400218967336992,312400218673012936,817056406224656,10927434466,4738
%N Multi-table menage numbers T(n,k) for n,k >= 1 equals the number of ways to seat the gentlemen from n*k married couples at n round tables with 2*k seats each such that (i) the gender of persons alternates around each table; and (ii) spouses do not sit next to each other; provided that the ladies are already properly seated (i.e., no two ladies sit next to each other).
%F T(n,k) = Sum_{j=0..n*k} (-1)^j * (n*k-j)! * [z^j] F(k,z)^n, where F(1,z) = 1+z and F(k,z) = ((1-sqrt(1+4*z))/2)^(2*k) + ((1+sqrt(1+4*z))/2)^(2*k) for k >= 2. [Corrected by _Pontus von Brömssen_, Jun 01 2022]
%F T(n,k) = A341439(n,n*k). - _Pontus von Brömssen_, May 31 2022
%e Table T(n,k):
%e n=1: 0, 0, 1, 2, ...
%e n=2: 1, 4, 82, 4740, ...
%e n=3: 2, 80, 43390, 59216648, ...
%e n=4: 9, 4752, 59216968, 2649391488016, ...
%e n=5: 44, 440192, 164806652728, 312400218967336992, ...
%e ...
%o (PARI) { A277256(n,k) = my(m,s,g); m=n*k; s=sqrt(1+4*x+O(x^(m+1))); g=if(k==1,1+z,((1-s)/2)^(2*k)+((1+s)/2)^(2*k))^n; sum(j=0,m,(-1)^j*polcoeff(g,j)*(m-j)!); }
%Y Cf. A000179 (row n=1), A000166 (column k=1), A000316 (column k=2), A277257, A277265, A341439.
%K nonn,tabl
%O 1,4
%A _Max Alekseyev_, Oct 07 2016
|
