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).

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

Links

Table of n, a(n) for n=1..36.

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 A102405 A363575

Adjacent sequences: A277253 A277254 A277255 * A277257 A277258 A277259

Keyword

nonn,tabl

Author

Max Alekseyev, Oct 07 2016

Status

approved