A159040 A triangle of polynomial coefficients: p(x,n)=Sum[x^i*If[i == Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= (less than or equal to) Floor[n/2], (-1)^i*A109128[n, i], -(-1)^(n - i)* A109128[n, i]]], {i, 0, n}]/(1 - x).

1, 1, 1, 1, -4, 1, 1, -6, -6, 1, 1, -8, 11, -8, 1, 1, -10, 19, 19, -10, 1, 1, -12, 29, -40, 29, -12, 1, 1, -14, 41, -70, -70, 41, -14, 1, 1, -16, 55, -112, 139, -112, 55, -16, 1, 1, -18, 71, -168, 251, 251, -168, 71, -18, 1, 1, -20, 89, -240, 419, -504, 419, -240, 89, -20, 1

Offset: 0

Author

Roger L. Bagula, Apr 03 2009

Keywords

• sign
• tabl
• uned

Comments

Row sums are:

{1, 2, -2, -10, -3, 20, -4, -84, -5, 274, -6,...}.

Examples

			{1},
{1, 1},
{1, -4, 1},
{1, -6, -6, 1},
{1, -8, 11, -8, 1},
{1, -10, 19, 19, -10, 1},
{1, -12, 29, -40, 29, -12, 1},
{1, -14, 41, -70, -70, 41, -14, 1},
{1, -16, 55, -112, 139, -112, 55, -16, 1},
{1, -18, 71, -168, 251, 251, -168, 71, -18, 1},
{1, -20, 89, -240, 419, -504, 419, -240, 89, -20, 1}
		

Crossrefs

A109128

Programs

Mathematica

Clear[A, p, n, i];
A[n_, 0] := 1;
A[n_, n_] := 1;
A[n_, k_] := A[n - 1, k - 1] + A[n - 1, k] + 1;
p[x_, n_] = Sum[x^i*If[i == Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= Floor[n/2], (-1)^i*A[n, i], -(-1)^(n - i)*A[n, i]]], {i, 0, n}]/(1 - x);
Table[CoefficientList[FullSimplify[p[x, n]], x], {n, 1, 11}];
Flatten[%]

Formula

p(x,n)=Sum[x^i*If[i == Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= (less than or equal to) Floor[n/2], (-1)^i*A109128[n, i], -(-1)^(n - i)* A109128[n, i]]], {i, 0, n}]/(1 - x);

t(n,m)=coefficients(p(x,n),x)