0,2

The Chebyshev polynomials of the second kind are defined by the recurrence relation: U(0,x) = 1; U(1,x) = 2x; U(n+1,x) = 2x*U(n,x) - U(n-1,x).

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 28 2008: (Start)

Triangle read by rows, unsigned = A000012 * A028297

Row sums of absolute values give the Pell series, A000129. (End)

The row sums are: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,...}.

Wikipedia, (Chebyshev polynomials).

Tracale Austin, Hans Bantilan, Isao Jonas and Paul Kory, The Pfaffian Transformation, Journal of Integer Sequences, Vol. 11 (2008), page 25 [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 19 2008]

A generating function for U(n) is 1/(1 - 2tx + t^2). Given A038207, shift down columns to allow for (1, 1, 2, 2, 3, 3,...) terms in each row, then insert alternate signs.

t(n,m) = (-1)^m*Binomial[n - m, m]*2^(n - 2*m) [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 19 2008]

The first few Chebyshev polynomials of the second kind are:

1;

2x;

4x^2 - 1;

8x^3 - 4x;

16x^4 - 12x^2 + 1;

32x^5 - 32x^3 + 6x;

64x^6 - 80x^4 + 24x^2 - 1;

128x^7 - 192x^5 + 80x^3 - 8x;

256x^8 - 448x^6 + 240x^4 - 40x^2 + 1;

512x^9 - 1024x^7 _ 672x^5 - 160x^3 + 10x;

...

Contribution from Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 19 2008: (Start)

{1},

{2},

{4, -1},

{8, -4},

{16, -12, 1},

{32, -32, 6},

{64, -80, 24, -1},

{128, -192, 80, -8},

{256, -448, 240, -40, 1},

{512, -1024, 672, -160,10},

{1024, -2304, 1792, -560, 60, -1} (End)

Contribution from Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 19 2008: (Start)

t[n_, m_] = (-1)^m*Binomial[n - m, m]*2^(n - 2*m);

Table[Table[t[n, m], {m, 0, Floor[n/2]}], {n, 0, 10}];

Flatten[%] (End)

Cf. A038207, A053117.

A018297, A000129 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 28 2008]

Cf. A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851, [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 12 2009]

Sequence in context: A121685 A125810 A152195 this_sequence A127529 A091977 A112829

Adjacent sequences: A133153 A133154 A133155 this_sequence A133157 A133158 A133159

tabf,sign

Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 16 2007

More terms from Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 12 2009