

A081277


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


20



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
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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 nth row of the triangle (followed by zeros) = nth row of the A142978 array and nth 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 TwoToned Tilings, arXiv:2001.11089 [math.CO], 2020.


FORMULA

T(n, k) = (n+2k)*binomial(n+k1, k1)*2^(n1)/k, k > 0.
T(n, 0) defined by g.f. (1x)/(12x). Other rows are defined by (1x)/(12x)^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, k1) + 2*T(n1, 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 selffusion 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



