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%I #18 Nov 14 2013 12:04:59
%S 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,
%T 96,1,21,105,385,915,1479,1459,675,1,28,196,952,3130,7324,11692,11584,
%U 5413,1,36,336,2100,9090,28764,65784,104364,103605,48800
%N Triangle giving coefficients of ménage hit polynomials.
%C 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
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 198.
%H T. D. Noe, <a href="/A058057/b058057.txt">Rows n = 0..100 of triangle, flattened</a>
%F G.f.: Sum(n!*(x*y)^n/(1+x*(y-1))^(2*n+1),n=0..infinity). [_Vladeta Jovovic_, Dec 13 2009]
%e 1; 1,0; 1,1,0; 1,3,1,1; 1,6,6,8,3; ...
%p 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);
%t 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_ *)
%Y Diagonals give A000271, A000426, A000222, A000386, A000450, A058085, A058086.
%Y Cf. A080018, A080061.
%K nonn,easy,nice,tabl
%O 0,8
%A _N. J. A. Sloane_, Dec 02 2000
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