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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.

%I A080018 #21 Jan 06 2016 11:50:00
%S A080018 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,
%T A080018 13,206,775,1288,1407,830,437,76,21,1708,6140,10366,10384,7298,3100,
%U A080018 1138,152,34,15702,55427,91296,92896,63140,31278,10048,2744,294,55
%N 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.
%D A080018 J. Riordan, The enumeration of permutations with three-ply staircase restrictions, unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963.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
%H A080018 Alois P. Heinz, <a href="/A080018/b080018.txt">Rows n = 0..20, flattened</a>
%e A080018 1;
%e A080018 0,  1;
%e A080018 0,  0,  2;
%e A080018 0,  1,  2,  3;
%e A080018 1,  2, 10,  6,  5;
%e A080018 4, 20, 28, 44, 16, 8;
%e A080018 ...
%e A080018 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.
%p A080018 with(LinearAlgebra):
%p A080018 T:= proc(n) option remember; local p;
%p A080018       if n=0 then 1 else
%p A080018         p:= Permanent(Matrix(n, (i,j)-> `if`(abs(i-j)<2, x, 1)));
%p A080018         seq(coeff(p, x, i), i=0..n)
%p A080018       fi
%p A080018     end:
%p A080018 seq(T(n), n=0..10);  # _Alois P. Heinz_, Jul 03 2013
%t A080018 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_ *)
%Y A080018 Row sums = A000142, first column = A001883, second column = A001884, third column = A001885, fourth column = A001886.
%Y A080018 Main diagonal and lower diagonal give: A000045(n+1), A178523. - _Alois P. Heinz_, Jul 03 2013
%K A080018 nonn,tabl
%O A080018 0,6
%A A080018 _Vladeta Jovovic_, _Vladimir Baltic_, Jan 20 2003