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A081277
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Square array of unsigned coefficients of Chebyshev polynomials of the first kind.
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19
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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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 DELEHAM (kolotoko(AT)wanadoo.fr), Aug 09 2005
Antidiagonal sums are in A025192 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 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 (qntmpkt(AT)yahoo.com), 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. [From Clark Kimberling, Aug 4 2011]
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FORMULA
| T(n, k) := (n+2k)C(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 DELEHAM (kolotoko(AT)wanadoo.fr), Aug 12 2005
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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
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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 4 2011 *)
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CROSSREFS
| Cf. A079628.
Cf. A142978, A104698.
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 23 2009: (Start)
Cf. A167580 and A167591.
(End)
Sequence in context: A092879 A073370 A129675 * A079628 A140287 A077951
Adjacent sequences: A081274 A081275 A081276 * A081278 A081279 A081280
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Mar 16 2003
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