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%I #33 May 08 2018 15:11:54
%S 1,1,1,1,3,1,3,4,1,10,5,1,10,15,6,1,35,21,7,1,35,56,28,8,1,126,84,36,
%T 9,1,126,210,120,45,10,1,462,330,165,55,11,1,462,792,495,220,66,12,1,
%U 1716,1287,715,286,78,13,1,1716,3003,2002,1001,364,91,14,1,6435,5005,3003
%N Triangle of expansions of powers of x in terms of Chebyshev polynomials T_n (x).
%C This triangle is the right half of Pascal's triangle (A007318), but with each number along the center of Pascal's triangle (except the 1 at the top) divided by 2. - Benjamin Schak (schak(AT)math.upenn.edu), Dec 02 2005
%C For n>=2 found in A002378, a(n)=A034869(n)/2, for all others a(n)=A034869(n). - _R. J. Mathar_, May 13 2006
%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.
%H Vincenzo Librandi, <a href="/A008311/b008311.txt">Table of n, a(n) for n = 0..5775</a>
%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 H. J. Brothers, <a href="http://www.brotherstechnology.com/docs/Pascal's_Prism_(supplement).pdf">Pascal's Prism: Supplementary Material</a>.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F Sum_{k, 0<=k}T(n,k)*cos(kx) = 2^(n-1)*cos(x)^n. - _Philippe Deléham_, Mar 09 2013
%e Triangle begins:
%e 1;
%e -, 1;
%e 1, -, 1;
%e -, 3, -, 1;
%e 3, -, 4, -, 1;
%e -, 10, -, 5, -, 1;
%e ...
%e From _Philippe Deléham_, Mar 09 2013: (Start)
%e cos(x) = 1*cos(x),
%e 2*cos(x)^2 = 1 + cos(2x),
%e 4*cos(x)^3 = 3*cos(x) + cos(3x),
%e 8*cos(x)^4 = 3 + 4*cos(2x) + cos(4x),
%e 16*cos(x)^5 = 10*cos(x) + 5*cos(3x) + cos(5x), etc. (End)
%p printf("1,") ; for n from 1 to 20 do for j from n mod 2 to n by 2 do if j = 0 then printf("%d,",binomial(n,(n-j)/2)/2) ; else printf("%d,",binomial(n,(n-j)/2)) ; fi ; od ; od ; # _R. J. Mathar_, May 13 2006
%t row[n_] := If[n == 0, {1}, Table[If[j == 0, Binomial[n, (n - j)/2]/2, Binomial[n, (n - j)/2]], {j, Mod[n, 2], n, 2}]];
%t Table[row[n], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, May 05 2017, after _R. J. Mathar_ *)
%Y With zeros: A100257.
%K nonn,tabf,easy
%O 0,5
%A _N. J. A. Sloane_
%E Corrected by _Philippe Deléham_, Nov 12 2005
%E More terms from _R. J. Mathar_, May 13 2006
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