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A058087 Triangle giving coefficients of ménage hit polynomials. 10
1, 2, 0, 2, 0, 0, 2, 3, 0, 1, 2, 8, 4, 8, 2, 2, 15, 20, 40, 30, 13, 2, 24, 60, 152, 210, 192, 80, 2, 35, 140, 469, 994, 1477, 1344, 579, 2, 48, 280, 1232, 3660, 7888, 11672, 10800, 4738, 2, 63, 504, 2856, 11268, 32958, 70152, 104256, 97434, 43387 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

I. Kaplansky and J. Riordan, The probleme des menages, Scripta Mathematica, 1946, 12 (2), 113-124.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 198.

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. - N. J. A. Sloane, Jul 06 2014

LINKS

Table of n, a(n) for n=0..54.

Anthony C. Robin, 90.72 Circular Wife Swapping, The Mathematical Gazette, Vol. 90, No. 519 (Nov., 2006), pp. 471-478.

FORMULA

G.f.: x*y+(1-x*(y-1))*Sum(n!*(x*y)^n/(1+x*(y-1))^(2*n+1),n=0..infinity). [Vladeta Jovovic, Dec 14 2009]

EXAMPLE

1;

2,0;

2,0,0;

2,3,0,1;

2,8,4,8,2;

2,15,20,40,30,13;

2,24,60,152,210,192,80;

2,35,140,469,994,1477,1344,579;

2,48,280,1232,3660,7888,11672,10800,4738;

2,63,504,2856,11268,32958,70152,104256,97434...

MAPLE

U := proc(n) local k; add( (2*n/(2*n-k))*binomial(2*n-k, k)*(n-k)!*(x-1)^k, k=0..n); end; W := proc(r, s) coeff( U(r), x, s ); end; a := (n, k)->W(n, n-k); # valid for n >= 2.

MATHEMATICA

u[n_] := Sum[ 2*n/(2*n-k)*Binomial[2*n-k, k]*(n-k)!*(x-1)^k, {k, 0, n}]; w[r_, s_] := Coefficient[u[r], x, s]; a[n_, k_] := w[n, n-k]; a[0, 0]=1; a[1, 1]=0; Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 10 2012, translated from Maple *)

CROSSREFS

Diagonals give A000179, A000425, A000033, A000159, A000181, A000185, A058089, A058090.

Essentially a mirror image of A094314.

Sequence in context: A216176 A128765 A193511 * A254218 A263147 A298100

Adjacent sequences:  A058084 A058085 A058086 * A058088 A058089 A058090

KEYWORD

nonn,easy,tabl,nice

AUTHOR

N. J. A. Sloane, Dec 02 2000

STATUS

approved

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Last modified October 17 12:01 EDT 2018. Contains 316279 sequences. (Running on oeis4.)