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A277256
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).
6
0, 1, 0, 2, 4, 1, 9, 80, 82, 2, 44, 4752, 43390, 4740, 13, 265, 440192, 59216968, 59216648, 439794, 80, 1854, 59245120, 164806652728, 2649391488016, 164806435822, 59216644, 579, 14833, 10930514688, 817056761525488, 312400218967336992, 312400218673012936, 817056406224656, 10927434466, 4738
OFFSET
1,4
FORMULA
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]
T(n,k) = A341439(n,n*k). - Pontus von Brömssen, May 31 2022
EXAMPLE
Table T(n,k):
n=1: 0, 0, 1, 2, ...
n=2: 1, 4, 82, 4740, ...
n=3: 2, 80, 43390, 59216648, ...
n=4: 9, 4752, 59216968, 2649391488016, ...
n=5: 44, 440192, 164806652728, 312400218967336992, ...
...
PROG
(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)!); }
CROSSREFS
Cf. A000179 (row n=1), A000166 (column k=1), A000316 (column k=2), A277257, A277265, A341439.
Sequence in context: A240717 A166900 A192437 * A208936 A373756 A102405
KEYWORD
nonn,tabl
AUTHOR
Max Alekseyev, Oct 07 2016
STATUS
approved