A080061 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<=2 else m(i,j)=1.

1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 1, 4, 8, 10, 1, 5, 21, 38, 34, 21, 1, 33, 122, 209, 206, 109, 40, 1, 236, 849, 1400, 1351, 836, 295, 72, 1, 1918, 6719, 10849, 10543, 6629, 2821, 715, 125, 1, 17440, 59873, 95516, 92708, 60284, 26870, 8372, 1604, 212, 1, 175649, 593686

Offset: 0

Author

Vladeta Jovovic, Vladimir Baltic, Jan 23 2003

Keywords

• nonn
• tabl

Examples

			1;
0,1;
0,1,1;
0,1,4,1;
1,4,8,10,1;
5,21,38,34,21,1;
... P(5; x) = Permanent(Matrix(5, 5, [[x,1,1,1,1],[x,x,1,1,1],[x,x,x,1,1],[1,x,x,x,1],[1,1,x,x,x]]))= 5+21*x+38*x^2+34*x^3+21*x^4+x^5.
		

References

• J. Riordan, The enumeration of permutations with three-ply staircase restrictions, unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963. See Table 1. - N. J. A. Sloane, Aug 27 2013 (See A001883)

Links

• R. J. Mathar, Table of n, a(n) for n = 0..209

Crossrefs

Row sums = A000142, first column = A001887, second column = A001888, third column = A001889, fourth column = A001890, A080018.

Cf. A001883.

Programs

Maple

A080061_line := proc(n)
    local M,r,c,p,pord ;
    if n = 0 then
        return [1] ;
    else
        M := Matrix(n,n) ;
        for r to n do
        for c to n do
            if r-c >=0 and r-c <=2 then
                M[r,c] := x ;
            else
                M[r,c] := 1 ;
            end if;
        end do:
        end do:
        p := LinearAlgebra[Permanent](M) ;
        pord := degree(p) ;
        [seq( coeff(p,x,r),r=0..pord)] ;
    end if;
end proc:
for n from 0 to 10 do
    print(A080061_line(n)) ;
end do: # R. J. Mathar, Sep 18 2013

Mathematica

M[n_] := Table[If[0 <= i-j <= 2, x, 1], {i, 1, n}, {j, 1, n}]; M[0]={{1}}; Table[CoefficientList[Permanent[M[n]], x], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 06 2016 *)