A143080 Triangular sequence of coefficients from an exponential based polynomial: p(x,n)=If[n == 0, 1, -(2*n - 1)!*x^(2*n)*(Sum[x^i/(i + 2*n)!, {i, 0, Infinity}] - Exp[x]/x^(2*n))].

1, 1, 1, 6, 6, 3, 1, 120, 120, 60, 20, 5, 1, 5040, 5040, 2520, 840, 210, 42, 7, 1, 362880, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1, 39916800, 39916800, 19958400, 6652800, 1663200, 332640, 55440, 7920, 990, 110, 11, 1, 6227020800, 6227020800

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Oct 15 2008

Keywords

• nonn
• tabf

Comments

Row sums are: 1, 2, 16, 326, 13700, 986410, 108505112, ...

These polynomials are based on: f(x)=1/(1-x)-exp(x).

The n-th row is the coefficient list of the permanental polynomial of the (2n-1)X(2n-1) matrix consisting entirely of 1's (see latter Mathematica code below). - John M. Campbell, Jul 05 2012

Examples

			{1},
{1, 1},
{6, 6, 3, 1},
{120, 120, 60, 20, 5, 1},
{5040, 5040, 2520, 840, 210, 42, 7, 1},
{362880, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1},
{39916800, 39916800, 19958400, 6652800, 1663200, 332640, 55440, 7920, 990, 110, 11, 1},
{6227020800, 6227020800, 3113510400, 1037836800, 259459200, 51891840, 8648640, 1235520, 154440, 17160, 1716, 156, 13, 1}
		

Crossrefs

Cf. A000522, A094587.

Programs

Mathematica

Clear[f, x, n, a]; f[x_, n_] := f[x, n] = If[n == 0, 1, -(2*n - 1)!*x^(2*n)*(Sum[x^i/(i + 2*n)!, {i, 0, Infinity}] - Exp[x]/x^(2*n))]; a = Table[CoefficientList[FullSimplify[f[x, n]], x], {n, 0, 10}]; Flatten[a]
Permanent[m_List] := With[{v = Array[x, Length[m]]}, Coefficient[Times @@ (m.v), Times @@ v]]; A[q_] := Array[1 &, {q, q}]; Flatten[Table[Abs[CoefficientList[Expand[Permanent[A[2*n-1] - IdentityMatrix[2*n-1]*x]], x]], {n, 6}]] (* John M. Campbell, Jul 05 2012 *)

Formula

p(x,n)=If[n == 0, 1, -(2*n - 1)!*x^(2*n)*(Sum[x^i/(i + 2*n)!, {i, 0, Infinity}] - Exp[x]/x^(2*n))]; t(n,m)=Coefficients(p)x,n)).