A080018 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 -1<=i-j<=1 else m(i,j)=1.

1, 0, 1, 0, 0, 2, 0, 1, 2, 3, 1, 2, 10, 6, 5, 4, 20, 28, 44, 16, 8, 29, 104, 207, 180, 151, 36, 13, 206, 775, 1288, 1407, 830, 437, 76, 21, 1708, 6140, 10366, 10384, 7298, 3100, 1138, 152, 34, 15702, 55427, 91296, 92896, 63140, 31278, 10048, 2744, 294, 55

Offset: 0

Author

Vladeta Jovovic, Vladimir Baltic, Jan 20 2003

Keywords

• nonn
• tabl

Examples

			1;
0,  1;
0,  0,  2;
0,  1,  2,  3;
1,  2, 10,  6,  5;
4, 20, 28, 44, 16, 8;
...
P(4; x) = Permanent(MATRIX([[x, x, 1, 1], [x, x, x, 1], [1, x, x, x], [1, 1, x, x]])) = 1+2*x+10*x^2+6*x^3+5*x^4.
		

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• J. Riordan, The enumeration of permutations with three-ply staircase restrictions, unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963.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

Links

• Alois P. Heinz, Rows n = 0..20, flattened

Crossrefs

Row sums = A000142, first column = A001883, second column = A001884, third column = A001885, fourth column = A001886.

Main diagonal and lower diagonal give: A000045(n+1), A178523. - Alois P. Heinz, Jul 03 2013

Programs

Maple

with(LinearAlgebra):
T:= proc(n) option remember; local p;
      if n=0 then 1 else
        p:= Permanent(Matrix(n, (i,j)-> `if`(abs(i-j)<2, x, 1)));
        seq(coeff(p, x, i), i=0..n)
      fi
    end:
seq(T(n), n=0..10);  # Alois P. Heinz, Jul 03 2013

Mathematica

t[0] = {1}; t[n_] := CoefficientList[Permanent[Array[If[Abs[#1 - #2] < 2, x, 1]&, {n, n}]], x]; Table[t[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)