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%I #53 Aug 12 2022 21:15:54
%S 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,
%T 126,1,10,45,120,210,126,1,11,55,165,330,462,1,12,66,220,495,792,462,
%U 1,13,78,286,715,1287,1716,1,14,91,364,1001,2002,3003,1716,1,15,105,455,1365,3003,5005
%N Irregular triangle read by rows: one half of the coefficients of the expansion of (2*x)^n in terms of Chebyshev T-polynomials.
%C The entry a(0,0) should actually be 1/2.
%C The row lengths of this array are [1,1,2,2,3,3,...] = A004526.
%C 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
%C 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
%D 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.
%D 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.
%H Robert Israel, <a href="/A008314/b008314.txt">Table of n, a(n) for n = 0..10099</a> (rows 0 to 199, flattened)
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Suyoung Choi and Hanchul Park, <a href="http://arxiv.org/abs/1210.3776">A new graph invariant arises in toric topology</a>, arXiv preprint arXiv:1210.3776 [math.AT], 2012.
%H Mathematics Stack Exchange, <a href="http://math.stackexchange.com/questions/345529">Bijection between number of partitions of 2n satisfying certain conditions with number of partitions of n</a>, April-March 2013.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F 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
%e [1/2], [1], [1,2/2=1], [1,3], [1,4,6/2=3], [1,5,10], [1,6,15,20/2=10],...
%e From _Wolfdieter Lang_, Aug 01 2014: (Start)
%e This irregular triangle begins (even n has falling even T-polynomial indices, odd n has falling odd T-indices):
%e n\k 1 2 3 4 5 6 7 8 ...
%e 0: 1/2 (but a(0,1) = 1)
%e 1: 1
%e 2: 1 1
%e 3: 1 3
%e 4: 1 4 3
%e 5: 1 5 10
%e 6: 1 6 15 10
%e 7: 1 7 21 35
%e 8: 1 8 28 56 35
%e 9: 1 9 36 84 126
%e 10: 1 10 45 120 210 126
%e 11: 1 11 55 165 330 462
%e 12: 1 12 66 220 495 792 462
%e 13: 1 13 78 286 715 1287 1716
%e 14: 1 14 91 364 1001 2002 3003 1716
%e 15: 1 15 105 455 1365 3003 5005 6435
%e ...
%e (2*x)^5 = 2*(1*T_5(x) + 5*T_3(x) + 10*T_1(x)),
%e (2*x)^6 = 2*(1*T_6(x) + 6*T_4(x) + 15*T_3(x) + 10*T_0(x)).
%e (End)
%p F:= proc(n) local q;
%p q:= combine(2^(n-1)*cos(t)^n,trig);
%p if n::even then
%p seq(coeff(q,cos((n-2*j)*t)),j=0..n/2-1),eval(q,cos=0)
%p else
%p seq(coeff(q,cos((n-2*j)*t)),j=0..(n-1)/2)
%p fi
%p end proc:
%p 1, seq(F(n),n=1..15); # _Robert Israel_, Jul 25 2016
%t 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}];
%t (* or with combinatorics *)
%t 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}];
%t 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 *)
%Y Cf. A007318, A008311.
%Y Bisection triangles: A122366 (odd numbered rows), A127673 (even numbered rows).
%K nonn,tabf
%O 0,6
%A _N. J. A. Sloane_
%E Name reformulated by _Wolfdieter Lang_, Aug 01 2014
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