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%I #35 Jul 26 2022 16:29:41
%S 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,
%T 32,1,13,72,220,400,432,256,64,1,15,98,364,840,1232,1120,576,128,1,17,
%U 128,560,1568,2912,3584,2816,1280,256,1,19,162,816,2688,6048,9408,9984,6912
%N Square array of unsigned coefficients of Chebyshev polynomials of the first kind.
%C Rows include A011782, A001792, A001793, A001794, A006974.
%C 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
%C Antidiagonal sums are in A025192. - _Philippe Deléham_, Dec 04 2006
%C 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
%C 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
%H Robert Davis, Greg Simay, <a href="https://arxiv.org/abs/2001.11089">Further Combinatorics and Applications of Two-Toned Tilings</a>, arXiv:2001.11089 [math.CO], 2020.
%F T(n, k) = (n+2k)*binomial(n+k-1, k-1)*2^(n-1)/k, k > 0.
%F T(n, 0) defined by g.f. (1-x)/(1-2x). Other rows are defined by (1-x)/(1-2x)^n.
%F 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
%F G.f. of the triangular interpretation: (-1+x*y)/(-1+2*x*y+x). - _R. J. Mathar_, Aug 11 2015
%e Rows begin
%e 1, 1, 2, 4, 8, ...
%e 1, 3, 8, 20, 48, ...
%e 1, 5, 18, 56, 160, ...
%e 1, 7, 32, 120, 400, ...
%e 1, 9, 50, 220, 840, ...
%e ...
%e As a triangle:
%e 1;
%e 1, 1;
%e 1, 3, 2;
%e 1, 5, 8, 4;
%e 1, 7, 18, 20, 8;
%t (* Program generates triangle A081277 as the self-fusion of Pascal's triangle *)
%t z = 8; a = 1; b = 1; c = 1; d = 1;
%t p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
%t t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t g[n_] := CoefficientList[w[n, x], {x}]
%t TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A081277 *)
%t TableForm[Table[g[n], {n, -1, z}]]
%t Flatten[Table[g[n], {n, -1, z}]] (* abs val of A118800 *)
%t Factor[w[6, x]]
%t (* _Clark Kimberling_, Aug 04 2011 *)
%Y Cf. A079628.
%Y Cf. A142978, A104698.
%Y Cf. A167580 and A167591. - _Johannes W. Meijer_, Nov 23 2009
%Y Cf. A053120 (antidiagonals give signed version) and A124182 (skewed version). - _Mathias Zechmeister_, Jul 26 2022
%K easy,nonn,tabl
%O 0,5
%A _Paul Barry_, Mar 16 2003
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