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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). 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

Table of n, a(n) for n=1..34.

FORMULA

T(n,k) = Sum_{j=0..n*k} (-1)^j * (n-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.

EXAMPLE

Table T(n,k):

n=1: 0, 0, 1, 2, 13, 80, ...

n=2: 1, 4, 82, 4740, 439794, ...

n=3: 2, 80, 43390, 59216648, ...

n=4: 9, 4752, 59216968, ...

n=5: 44, 440192, 164806652728, ...

...

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.

Sequence in context: A240717 A166900 A192437 * A208936 A102405 A271206

Adjacent sequences:  A277253 A277254 A277255 * A277257 A277258 A277259

KEYWORD

nonn,tabl

AUTHOR

Max Alekseyev, Oct 07 2016

STATUS

approved

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Last modified May 17 21:11 EDT 2021. Contains 343990 sequences. (Running on oeis4.)