A142701 A triangular sequence of coefficients made from a product sum of the Pascal/binomial and the Chebyshev T Polynomials: t(n,m)=-Sum[Binomial[n + 1, k + 1]*CoefficientList[ChebyshevT[k + 1, x], x][[m]], {k, m, n}].

1, 3, 3, 5, 12, 8, 5, 25, 40, 20, 1, 30, 102, 120, 48, -7, 7, 154, 364, 336, 112, -15, -56, 88, 672, 1184, 896, 256, -15, -135, -216, 624, 2592, 3600, 2304, 576, 1, -150, -710, -480, 3280, 9120, 10400, 5760, 1280, 33, 11, -946, -2860, 176, 14432, 29920, 28864, 14080, 2816, 65, 396, -60, -4752, -9288, 8448, 56384

Offset

1,2

Comments

Roe sums are:

{1, 6, 25, 90, 301, 966, 3025, 9330, 28501, 86526, 261625, 788970}.

The resulting coefficients are the Chebyshev orthogonal base

for Pascal's triangle.

Links

Table of n, a(n) for n=1..62.

Formula

t(n,m)=-Sum[Binomial[n + 1, k + 1]*CoefficientList[ChebyshevT[k + 1, x], x][[m]], {k, m, n}].

Example

{1},
{3, 3},
{5, 12, 8},
{5, 25, 40, 20},
{1, 30, 102, 120, 48},
{-7, 7, 154, 364, 336, 112},
{-15, -56, 88, 672, 1184, 896, 256},
{-15, -135, -216, 624, 2592, 3600, 2304, 576},
{1, -150, -710, -480, 3280, 9120, 10400, 5760, 1280},
{33, 11, -946, -2860, 176, 14432, 29920,28864, 14080,2816},
{65, 396, -60, -4752, -9288, 8448, 56384, 92928, 77568, 33792, 6144},
{65, 845, 2652, -1352, -20072, -23504, 55744, 202176, 276224, 203008, 79872, 13312}

Mathematica

Clear[t, n, m, k]; t[n_, m_] := -Sum[ Binomial[n + 1, k + 1]*CoefficientList[ChebyshevT[k + 1, x], x][[ m]], {k, m, n}]; Table[Table[t[n, m], {m, 1, n}], {n, 1, 12}]; Flatten[%]

Crossrefs

Sequence in context: A187874 A126318 A360024 * A257351 A227616 A079439

Adjacent sequences: A142698 A142699 A142700 * A142702 A142703 A142704

Keyword

uned,sign

Author

Roger L. Bagula and Gary W. Adamson, Sep 23 2008

Status

approved