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A008314 Irregular triangle read by rows: one half of the coefficients of the expansion of (2*x)^n in terms of Chebyshev T-polynomials. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
COMMENTS
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
REFERENCES
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.
LINKS
Robert Israel, Table of n, a(n) for n = 0..10099 (rows 0 to 199, flattened)
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].
Suyoung Choi and Hanchul Park, A new graph invariant arises in toric topology, arXiv preprint arXiv:1210.3776 [math.AT], 2012.
FORMULA
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
EXAMPLE
[1/2], [1], [1,2/2=1], [1,3], [1,4,6/2=3], [1,5,10], [1,6,15,20/2=10],...
From Wolfdieter Lang, Aug 01 2014: (Start)
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)
MAPLE
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:
1, seq(F(n), n=1..15); # Robert Israel, Jul 25 2016
MATHEMATICA
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 *)
CROSSREFS
Bisection triangles: A122366 (odd numbered rows), A127673 (even numbered rows).
Sequence in context: A101038 A064883 A090844 * A104568 A030758 A347065
KEYWORD
nonn,tabf
AUTHOR
EXTENSIONS
Name reformulated by Wolfdieter Lang, Aug 01 2014
STATUS
approved

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