A101124 Number triangle associated to Chebyshev polynomials of first kind.

1, 0, 1, -1, 1, 1, 0, 1, 2, 1, 1, 1, 7, 3, 1, 0, 1, 26, 17, 4, 1, -1, 1, 97, 99, 31, 5, 1, 0, 1, 362, 577, 244, 49, 6, 1, 1, 1, 1351, 3363, 1921, 485, 71, 7, 1, 0, 1, 5042, 19601, 15124, 4801, 846, 97, 8, 1, -1, 1, 18817, 114243, 119071, 47525, 10081, 1351, 127, 9, 1, 0, 1, 70226, 665857, 937444, 470449, 120126, 18817, 2024, 161

Offset

0,9

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Index entries for sequences related to Chebyshev polynomials.

Formula

Number triangle S(n, k)=T(n-k, k), k<n, S(n, n)=1, 0 otherwise, where T(n, k)=(n/2)sum{j=0..floor(n/2), C(n-j, j)(-1)^j*(2k)^(n-2j)}.

Columns have g.f. x^k(1-kx)/(1-2kx+x^2).

Also, square array if(n=0, 1, T(n, k)) read by antidiagonals.

Example

As a number triangle, rows begin:
  {1},
  {0,1},
  {-1,1,1},
  {0,1,2,1},
  ...
As a square array, rows begin
   1, 1,  1,   1,    1, ...
   0, 1,  2,   3,    4, ...
  -1, 1,  7,  17,   31, ...
   0, 1, 26,  99,  244, ...
   1, 1, 97, 577, 1921, ...

Mathematica

T[n_, k_] := SeriesCoefficient[x^k (1 - k x)/(1 - 2 k x + x^2), {x, 0, n}];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 12 2017 *)

Crossrefs

Columns include A001075, A001541, A001091, A001079, A023038, A011943.

Row sums are A101125.

Diagonal sums are A101126.

Main diagonal gives A115066.

Mirror of A322836.

Cf. A053120.

Sequence in context: A137296 A329291 A294105 * A011127 A172970 A172971

Adjacent sequences: A101121 A101122 A101123 * A101125 A101126 A101127

Keyword

easy,sign,tabl

Author

Paul Barry, Dec 02 2004

Status

approved