A173293 Antidiagonal expansion of rational polynomial with factors: p(x,n) = If[n == 0, 1/(1 - x), x*ChebyshevU[n, x]/ChebyshevT[n + 1, x]].

1, 1, 0, 1, 0, 1, 1, -16, 0, 0, 1, 0, -24, 0, 1, 1, -128, 0, -256, 0, 0, 1, 0, -288, 0, -200, 0, 1, 1, -1024, 0, -24576, 0, -1296, 0, 0, 1, 0, -3456, 0, -12000, 0, -784, 0, 1, 1, -8192, 0, -2621440, 0, -590976, 0, -4096, 0, 0

Offset

0,8

Comments

Row sums are {1, 1, 2, -15, -22, -383, -486, -26895, -16238, -3224703, ...}.

The rational function here is associated with tan(n*arctan(x)).

Links

Table of n, a(n) for n=0..54.

Formula

p(x,n) = If[n == 0, 1/(1 - x), x*ChebyshevU[n, x]/ChebyshevT[n + 1, x]];

a(n,m) = (n+1)^m*expansion(p(x,n));

t(n,m) = antidiagonal(t(n,m)).

Example

{1},
{1, 0},
{1, 0, 1},
{1, -16, 0, 0},
{1, 0, -24, 0, 1},
{1, -128, 0, -256, 0, 0},
{1, 0, -288, 0, -200, 0, 1},
{1, -1024, 0, -24576, 0, -1296, 0, 0},
{1, 0, -3456, 0, -12000, 0, -784, 0, 1},
{1, -8192, 0, -2621440, 0, -590976, 0, -4096, 0, 0}

Mathematica

p[x_, n_] = If[n == 0, 1/(1 - x), x*ChebyshevU[n, x]/ChebyshevT[n + 1, x]];
a = Table[Table[(n + 1)^(m + 1)*SeriesCoefficient[Series[p[x, n], {x, 0, 50}], m], { m, 0, 20}], {n, 0, 20}];
Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];
Flatten[%]

Crossrefs

Sequence in context: A336432 A169767 A225611 * A008433 A347803 A010111

Adjacent sequences: A173290 A173291 A173292 * A173294 A173295 A173296

Keyword

sign,tabl,uned

Author

Roger L. Bagula, Feb 15 2010

Status

approved