A081277 Square array of unsigned coefficients of Chebyshev polynomials of the first kind.

1, 1, 1, 1, 3, 2, 1, 5, 8, 4, 1, 7, 18, 20, 8, 1, 9, 32, 56, 48, 16, 1, 11, 50, 120, 160, 112, 32, 1, 13, 72, 220, 400, 432, 256, 64, 1, 15, 98, 364, 840, 1232, 1120, 576, 128, 1, 17, 128, 560, 1568, 2912, 3584, 2816, 1280, 256, 1, 19, 162, 816, 2688, 6048, 9408, 9984, 6912

Offset

0,5

Comments

Rows include A011782, A001792, A001793, A001794, A006974.

Formatted as a triangular array, this is [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...] (see construction in A084938 ). - Philippe Deléham, Aug 09 2005

Antidiagonal sums are in A025192. - Philippe Deléham, Dec 04 2006

Binomial transform of n-th row of the triangle (followed by zeros) = n-th row of the A142978 array and n-th column of triangle A104698. - Gary W. Adamson, Jul 17 2008

When formatted as a triangle, A038763=fusion of polynomial sequences (x+1)^n and (x+1)^n; see A193722 for the definition of fusion of two polynomial sequences or triangular arrays. Row n of A038763, as a triangle, consists of coefficients of the product (x+1)*(x+2)^n. - Clark Kimberling, Aug 04 2011

Links

Table of n, a(n) for n=0..63.

Robert Davis, Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.

Formula

T(n, k) = (n+2k)*binomial(n+k-1, k-1)*2^(n-1)/k, k > 0.

T(n, 0) defined by g.f. (1-x)/(1-2x). Other rows are defined by (1-x)/(1-2x)^n.

T(n, 0) = 0 if n < 0, T(0, k) = 0 if k < 0, T(0, 0) = T(1, 0) = 1, T(n, k) = T(n, k-1) + 2*T(n-1, k); for example, 160 = 48 + 2*56 for n = 4 and k = 2. -Philippe Deléham, Aug 12 2005

G.f. of the triangular interpretation: (-1+x*y)/(-1+2*x*y+x). - R. J. Mathar, Aug 11 2015

Example

Rows begin
  1, 1,  2,   4,   8, ...
  1, 3,  8,  20,  48, ...
  1, 5, 18,  56, 160, ...
  1, 7, 32, 120, 400, ...
  1, 9, 50, 220, 840, ...
  ...
As a triangle:
  1;
  1,  1;
  1,  3,  2;
  1,  5,  8,  4;
  1,  7, 18, 20,  8;

Mathematica

(* Program generates triangle A081277 as the self-fusion of Pascal's triangle *)
z = 8; a = 1; b = 1; c = 1; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A081277 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* abs val of A118800 *)
Factor[w[6, x]]
(* Clark Kimberling, Aug 04 2011 *)

Crossrefs

Cf. A079628.

Cf. A142978, A104698.

Cf. A167580 and A167591. - Johannes W. Meijer, Nov 23 2009

Cf. A053120 (antidiagonals give signed version) and A124182 (skewed version). - Mathias Zechmeister, Jul 26 2022

Sequence in context: A129675 A232206 A209559 * A079628 A140287 A077951

Adjacent sequences: A081274 A081275 A081276 * A081278 A081279 A081280

Keyword

easy,nonn,tabl

Author

Paul Barry, Mar 16 2003

Status

approved