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A008314
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Irregular triangle read by rows: one half of the coefficients of the expansion of (2*x)^n in terms of Chebyshev T-polynomials.
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6
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1, 1, 1, 1, 1, 3, 1, 4, 3, 1, 5, 10, 1, 6, 15, 10, 1, 7, 21, 35, 1, 8, 28, 56, 35, 1, 9, 36, 84, 126, 1, 10, 45, 120, 210, 126, 1, 11, 55, 165, 330, 462, 1, 12, 66, 220, 495, 792, 462, 1, 13, 78, 286, 715, 1287, 1716, 1, 14, 91, 364, 1001, 2002, 3003, 1716, 1, 15, 105, 455, 1365, 3003, 5005
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OFFSET
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0,6
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COMMENTS
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The entry a(0,0) should actually be 1/2.
The row lengths of this array are [1,1,2,2,3,3,...] = A004526.
Row k also counts the binary strings of length k that have 0, 2 up to 2*floor(k/2) 'unmatched symbols'. See contributions by Marc van Leeuwen at the Mathematics Stack Exchange link. - Wouter Meeussen, Apr 17 2013
For n >= 1, T(n,k) is the coefficient of cos((n-2k)x) in the expression for 2^(n-1)*cos(x)^n as a sum of cosines of multiples of x. It is binomial(n,k) if k < n/2, while T(n,n/2) = binomial(n,n/2)/2 if n is even. - Robert Israel, Jul 25 2016
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
T. J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2nd ed., Wiley, New York, 1990, pp. 54-55, Ex. 1.5.31.
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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a(n,k) are the M_3 multinomial numbers A036040 for the partitions with m = 1 and 2 parts (in Abramowitz-Stegun order). - Wolfdieter Lang, Aug 01 2014
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EXAMPLE
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[1/2], [1], [1,2/2=1], [1,3], [1,4,6/2=3], [1,5,10], [1,6,15,20/2=10],...
This irregular triangle begins (even n has falling even T-polynomial indices, odd n has falling odd T-indices):
n\k 1 2 3 4 5 6 7 8 ...
0: 1/2 (but a(0,1) = 1)
1: 1
2: 1 1
3: 1 3
4: 1 4 3
5: 1 5 10
6: 1 6 15 10
7: 1 7 21 35
8: 1 8 28 56 35
9: 1 9 36 84 126
10: 1 10 45 120 210 126
11: 1 11 55 165 330 462
12: 1 12 66 220 495 792 462
13: 1 13 78 286 715 1287 1716
14: 1 14 91 364 1001 2002 3003 1716
15: 1 15 105 455 1365 3003 5005 6435
...
(2*x)^5 = 2*(1*T_5(x) + 5*T_3(x) + 10*T_1(x)),
(2*x)^6 = 2*(1*T_6(x) + 6*T_4(x) + 15*T_3(x) + 10*T_0(x)).
(End)
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MAPLE
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F:= proc(n) local q;
q:= combine(2^(n-1)*cos(t)^n, trig);
if n::even then
seq(coeff(q, cos((n-2*j)*t)), j=0..n/2-1), eval(q, cos=0)
else
seq(coeff(q, cos((n-2*j)*t)), j=0..(n-1)/2)
fi
end proc:
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MATHEMATICA
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Table[(c/@ Range[n, 0, -2]) /. Flatten[Solve[Thread[CoefficientList[Expand[1/2*(2*x)^n -Sum[c[k] ChebyshevT[k, x], {k, 0, n}]], x]==0]]], {n, 16}];
(* or with combinatorics *)
match[li:{(1|-1)..}]:= Block[{it=li, rot=0}, While[Length[Union[Join[it, {"(", ")"}]]]>3, rot++; it=RotateRight[it //.{a___, 1, b___String, -1, c___} ->{a, "(", b, ")", c}]]; RotateLeft[it, rot] /. {(1|-1)->0, "("->1, ")"->-1}];
Table[Last/@ Sort@ Tally[Table[Tr[Abs@ match[-1+2*IntegerDigits[n, 2]]], {n, 2^(k-1), 2^k-1}]], {k, 1, 16}]; (* Wouter Meeussen, Apr 17 2013 *)
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CROSSREFS
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Bisection triangles: A122366 (odd numbered rows), A127673 (even numbered rows).
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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