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A232818 Triangle of coefficients of polynomials equal permanent of the n X n matrix [1,2,...,n; n*x+1, n*x+2, ..., n*x+n; ...; (n-1)*n*x+1, (n-1)*n*x+2, ...,(n-1)*n*x+n].

%I A232818 #42 Jul 02 2022 16:09:09
%S A232818 1,6,4,216,198,36,23040,24640,7200,576,5400000,6375000,2362500,328800,
%T A232818 14400,2351462400,2982873600,1285956000,238533120,19051200,518400,
%U A232818 1707698764800,2291162509440,1100516981760,245735819280,27025656000,1383117120,25401600
%N A232818 Triangle of coefficients of polynomials equal permanent of the n X n matrix [1,2,...,n; n*x+1, n*x+2, ..., n*x+n; ...; (n-1)*n*x+1, (n-1)*n*x+2, ...,(n-1)*n*x+n].
%C A232818 The degree of n-th polynomial is n-1.
%C A232818 Its leading coefficient is T(n,1) = n^n*(n-1)!^2*(n+1)/2. - _M. F. Hasler_, Dec 01 2013
%H A232818 G. C. Greubel, <a href="/A232818/b232818.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A232818 P_n(x) = (-1)^n * Sum_{k=0..n-1} c_k(n) * x^k, where c_k(n)= n^k * Stirling1(n,n-k) * Stirling1(n+1,k+1) * (n-k)! * k!.
%F A232818 P_n(1) = A232773; P_n(0) = n!^2, P_n(1/n) = A204248(n) is permanent of n X n Toeplitz matrix with the first row n,n-1,...,1 (see our comment in A204248).
%e A232818                                    1
%e A232818                            6*x +   4
%e A232818               216*x^2 +  198*x +  36
%e A232818 23040*x^3 + 24640*x^2 + 7200*x + 576
%e A232818               ......
%t A232818 p[n_,x_]:=(-1)^n Sum[n^k x^k StirlingS1[n,n-k]StirlingS1[n+1,k+1](n-k)!k!,{k,0,n-1}];Flatten[Table[Reverse[CoefficientList[p[n,x],x]],{n,8}]] (* _Peter J. C. Moses_, Nov 30 2013 *)
%o A232818 (PARI) P(n)=(-1)^n*sum(k=0,n-1,n^k*x^k*stirling(n,n-k)*stirling(n+1,k+1)*(n-k)!*k!)
%o A232818 apply(t->Vec(t),vector(7,n,P(n))) /* _M. F. Hasler_, Dec 01 2013 */
%Y A232818 Cf. A232773, A204248, A094638.
%K A232818 nonn,tabl
%O A232818 1,2
%A A232818 _Vladimir Shevelev_, Nov 30 2013
%E A232818 More terms from _Peter J. C. Moses_, Nov 30 2013