A144398 Coefficients of a symmetrical polynomial set:( Pascal's triangle with central zeros) p(x,n)=If[n <= 4, Sum[Binomial[n, i]*x^i, {i, 0, n}], x^n + n*x^(n - 1) + Binomial[n, 2]*x^(n - 2) + n*x + Binomial[n, 2]*x^2 + 1].

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 0, 15, 6, 1, 1, 7, 21, 0, 0, 21, 7, 1, 1, 8, 28, 0, 0, 0, 28, 8, 1, 1, 9, 36, 0, 0, 0, 0, 36, 9, 1, 1, 10, 45, 0, 0, 0, 0, 0, 45, 10, 1

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Oct 03 2008

Keywords

• nonn
• uned

Comments

Row sums are: (related to A014206)

{1, 2, 4, 8, 16, 32, 44, 58, 74, 92, 112}

Examples

			{1},
{1, 1},
{1, 2, 1},
{1, 3, 3, 1},
{1, 4, 6, 4, 1},
{1, 5, 10, 10, 5, 1},
{1, 6, 15, 0, 15, 6, 1},
{1, 7, 21, 0, 0, 21, 7, 1},
{1, 8, 28, 0, 0, 0, 28, 8, 1},
{1, 9, 36, 0, 0, 0, 0, 36, 9, 1},
{1, 10, 45, 0, 0, 0, 0, 0, 45, 10, 1}
		

Programs

Mathematica

Clear[p, n]; p[x_, n_] = If[n <= 4, Sum[Binomial[n, i]*x^i, {i, 0, n}], x^n + n*x^(n - 1) + Binomial[n, 2]*x^(n - 2) + n*x + Binomial[n, 2]*x^2 + 1]; Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[%]

Formula

p(x,n)=If[n <= 4, Sum[Binomial[n, i]*x^i, {i, 0, n}], x^n + n*x^(n - 1) + Binomial[n, 2]*x^(n - 2) + n*x + Binomial[n, 2]*x^2 + 1]; t(n,m)=coefficients(p(x,n)).