A144395 A designed polynomial set that gives a {1,6,1} quadratic and gives a symmetrical triangle of coefficients: p(x,n)=If[n == 2, 1, ((x + 1)^n -If[n == 0, 1, x^n + (n - 1)*x^(n - 1) + (n - 1)*x + 1])/x],.

1, 1, 1, 1, 6, 1, 1, 10, 10, 1, 1, 15, 20, 15, 1, 1, 21, 35, 35, 21, 1, 1, 28, 56, 70, 56, 28, 1, 1, 36, 84, 126, 126, 84, 36, 1, 1, 45, 120, 210, 252, 210, 120, 45, 1, 1, 55, 165, 330, 462, 462, 330, 165, 55, 1, 1, 66, 220, 495, 792, 924, 792, 495, 220, 66, 1

Offset

1,5

Comments

Row sums are:{1, 2, 8, 22, 52, 114, 240, 494, 1004, 2026, 4072}.

Links

Table of n, a(n) for n=1..66.

Formula

p(x,n)=If[n == 2, 1, ((x + 1)^n -If[n == 0, 1, x^n + (n - 1)*x^(n - 1) + (n - 1)*x + 1])/x]; t(n,m)=coefficients(p(x,n)).

Example

{1},
{1, 1},
{1, 6, 1},
{1, 10, 10, 1},
{1, 15, 20, 15, 1},
{1, 21, 35, 35, 21, 1},
{1, 28, 56, 70, 56, 28, 1},
{1, 36, 84, 126, 126, 84, 36, 1},
{1, 45, 120, 210, 252, 210, 120, 45, 1},
{1, 55, 165, 330, 462, 462, 330, 165, 55, 1},
{1, 66, 220, 495, 792, 924, 792, 495, 220, 66, 1}

Mathematica

Clear[p, x, n] p[x_, n_] = If[ n == 2, 1, ((x + 1)^n - If[n == 0, 1, x^n + (n - 1)*x^(n - 1) + (n - 1)*x + 1])/x]; Table[ExpandAll[p[x, n]], {n, 2, 12}]; Table[CoefficientList[p[x, n], x], {n, 2, 12}]; Flatten[%]

Crossrefs

Sequence in context: A176151 A204001 A363291 * A046621 A046617 A131063

Adjacent sequences: A144392 A144393 A144394 * A144396 A144397 A144398

Keyword

nonn,uned

Author

Roger L. Bagula and Gary W. Adamson, Oct 02 2008

Status

approved