A058057 Triangle giving coefficients of ménage hit polynomials.

1, 1, 0, 1, 1, 0, 1, 3, 1, 1, 1, 6, 6, 8, 3, 1, 10, 20, 38, 35, 16, 1, 15, 50, 134, 213, 211, 96, 1, 21, 105, 385, 915, 1479, 1459, 675, 1, 28, 196, 952, 3130, 7324, 11692, 11584, 5413, 1, 36, 336, 2100, 9090, 28764, 65784, 104364, 103605, 48800

Offset

0,8

Comments

Triangle of coefficients of polynomials P(n; x) = Permanent(M), where M=[m(i,j)] is n X n matrix defined by m(i,j)=x if 0<=i-j<=1 else m(i,j)=1. - Vladeta Jovovic, Jan 23 2003

References

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

Links

T. D. Noe, Rows n = 0..100 of triangle, flattened

Formula

G.f.: Sum(n!*(x*y)^n/(1+x*(y-1))^(2*n+1),n=0..infinity). [Vladeta Jovovic, Dec 13 2009]

Example

1; 1,0; 1,1,0; 1,3,1,1; 1,6,6,8,3; ...

Maple

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

Mathematica

max = 9; f[x_, y_] := Sum[n!*((x*y)^n/(1 + x*(y-1))^(2*n+1)), {n, 0, max}]; Flatten[ MapIndexed[ Take[#1, #2[[1]]] & , CoefficientList[ Series[f[x, y], {x, 0, max}, {y, 0, max}], {x, y}]]] (*Jean-François Alcover, Jun 29 2012, after Vladeta Jovovic *)

Crossrefs

Diagonals give A000271, A000426, A000222, A000386, A000450, A058085, A058086.

Cf. A080018, A080061.

Sequence in context: A291722 A297672 A256973 * A124372 A126470 A179701

Adjacent sequences: A058054 A058055 A058056 * A058058 A058059 A058060

Keyword

nonn,easy,nice,tabl

Author

N. J. A. Sloane, Dec 02 2000

Status

approved