A232818 - OEIS

%I #42 Jul 02 2022 16:09:09

%S 1,6,4,216,198,36,23040,24640,7200,576,5400000,6375000,2362500,328800,

%T 14400,2351462400,2982873600,1285956000,238533120,19051200,518400,

%U 1707698764800,2291162509440,1100516981760,245735819280,27025656000,1383117120,25401600

%N 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 The degree of n-th polynomial is n-1.

%C Its leading coefficient is T(n,1) = n^n*(n-1)!^2*(n+1)/2. - _M. F. Hasler_, Dec 01 2013

%H G. C. Greubel, <a href="/A232818/b232818.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F 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 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                                    1

%e                            6*x +   4

%e               216*x^2 +  198*x +  36

%e 23040*x^3 + 24640*x^2 + 7200*x + 576

%e               ......

%t 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 (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 apply(t->Vec(t),vector(7,n,P(n))) /* _M. F. Hasler_, Dec 01 2013 */

%Y Cf. A232773, A204248, A094638.

%K nonn,tabl

%O 1,2

%A _Vladimir Shevelev_, Nov 30 2013

%E More terms from _Peter J. C. Moses_, Nov 30 2013