OFFSET
0,6
COMMENTS
The men and women alternate.
REFERENCES
I. Kaplansky and J. Riordan, The problème des ménages, Scripta Math. 12, (1946), 113-124. See Table 1.
Tolman, L. Kirk, "Extensions of derangements", Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing, Humboldt State University, Arcata, California, September 5-7, 1979. Vol. 26. Utilitas Mathematica Pub., 1980. See Table I.
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
I. Kaplansky and J. Riordan, The problème des ménages, Scripta Math. 12, (1946), 113-124. [Scan of annotated copy]
Anthony C. Robin, 90.72 Circular Wife Swapping, The Mathematical Gazette, Vol. 90, No. 519 (Nov., 2006), pp. 471-478.
L. Takacs, On the probleme des menages, Discr. Math. 36 (3) (1981) 289-297, Table 1.
FORMULA
Sum_{k=0..n} T(n,k) = n!.
T(n, k) = Sum_{j=0..n-k} (-1)^j*(2*n*(n-k-j)!/(2*n-k-j))*binomial(k+j, k) * binomial(2*n-k-j, k+j) for n > 1, T(0, 0) = T(1, 1) = 1, and T(1, 0) = 0. - G. C. Greubel, May 15 2021
EXAMPLE
Triangle begins:
1;
0, 1;
0, 0, 2;
1, 0, 3, 2;
2, 8, 4, 8, 2;
13, 30, 40, 20, 15, 2;
80, 192, 210, 152, 60, 24, 2;
579, 1344, 1477, 994, 469, 140, 35, 2;
4738, 10800, 11672, 7888, 3660, 1232, 280, 48, 2;
...
MATHEMATICA
T[n_, k_]:= If[n<2, (1+(-1)^(n-k))/2, Sum[(-1)^j*(2*n*(n-k-j)!/(2*n-k-j))* Binomial[k+j, k]*Binomial[2*n-k-j, k+j], {j, 0, n-k}]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 15 2021 *)
PROG
(Sage)
def A094314(n, k): return (1+(-1)^(n+k))/2 if (n<2) else sum( (-1)^j*(2*n*factorial(n-k-j)/(2*n-k-j))*binomial(k+j, k)*binomial(2*n-k-j, k+j) for j in (0..n-k) )
flatten([[A094314(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 15 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, based on a suggestion from Anthony C Robin, Jun 02 2004
STATUS
approved