A176389 A symmetrical triangle of polynomial coefficients:p(x,n)=Sum[(1 + Binomial[n, m]*x)^m*(1 - Binomial[n, m]*x)^(n - m) + (x + Binomial[n, m])^m*(x - Binomial[n, m])^(n - m), {m, 0, n}]

2, 2, 2, 1, 0, 1, 4, -12, -12, 4, 791, 0, -120, 0, 791, 6, 16260, -280, -280, 16260, 6, -41312053, 0, 364560, 0, 364560, 0, -41312053, 8, -3163111000, -5544, 7035000, 7035000, -5544, -3163111000, 8, 383801047294219, 0, -343384323744, 0

Offset

0,1

Comments

Row sums are:

{2, 4, 2, -16, 1462, 31972, -81894986, -6312163072, 766915589474990, 224370055225370444,

-1400713126150624370808722,...}.

Links

Table of n, a(n) for n=0..39.

Formula

p(x,n)=Sum[(1 + Binomial[n, m]*x)^m*(1 - Binomial[n, m]*x)^(n - m) + (x + Binomial[n, m])^m*(x - Binomial[n, m])^(n - m), {m, 0, n}];

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

Example

{2},
{2, 2},
{1, 0, 1},
{4, -12, -12, 4},
{791, 0, -120, 0, 791},
{6, 16260, -280, -280, 16260, 6},
{-41312053, 0, 364560, 0, 364560, 0, -41312053},
{8, -3163111000, -5544, 7035000, 7035000, -5544, -3163111000, 8},
{383801047294219, 0, -343384323744, 0, 263534040, 0, -343384323744, 0, 383801047294219},
{10, 112211416526896260, -102960, -26391294385680, 2380277592, 2380277592, -26391294385680, -102960, 112211416526896260, 10},
{-700414925359462889249761, 0, 58364418631721000040, 0, -2134481017154640, 0, -2134481017154640, 0, 58364418631721000040, 0, -700414925359462889249761}

Mathematica

p[x_, n_] = Sum[(1 + Binomial[n, m]*x)^m*(1 - Binomial[n, m]*x)^(n - m) + (x + Binomial[n, m])^m*(x - Binomial[n, m])^(n - m), {m, 0, n}];
Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[%]

Crossrefs

Sequence in context: A246575 A357563 A112215 * A076451 A230536 A306257

Adjacent sequences: A176386 A176387 A176388 * A176390 A176391 A176392

Keyword

sign,tabl,uned

Author

Roger L. Bagula, Apr 16 2010

Status

approved