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].

1, 6, 4, 216, 198, 36, 23040, 24640, 7200, 576, 5400000, 6375000, 2362500, 328800, 14400, 2351462400, 2982873600, 1285956000, 238533120, 19051200, 518400, 1707698764800, 2291162509440, 1100516981760, 245735819280, 27025656000, 1383117120, 25401600

Offset: 1

Author

Vladimir Shevelev, Nov 30 2013

Keywords

• nonn
• tabl

Comments

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

Its leading coefficient is T(n,1) = n^n*(n-1)!^2*(n+1)/2. - M. F. Hasler, Dec 01 2013

Examples

			                                   1
                           6*x +   4
              216*x^2 +  198*x +  36
23040*x^3 + 24640*x^2 + 7200*x + 576
              ......
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A232773, A204248, A094638.

Programs

Mathematica

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 *)

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!)
apply(t->Vec(t),vector(7,n,P(n))) /* M. F. Hasler, Dec 01 2013 */

Formula

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!.

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).

Extensions

More terms from Peter J. C. Moses, Nov 30 2013