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A008311 Triangle of expansions of powers of x in terms of Chebyshev polynomials T_n (x).

%I A008311 #33 May 08 2018 15:11:54
%S A008311 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 A008311 9,1,126,210,120,45,10,1,462,330,165,55,11,1,462,792,495,220,66,12,1,
%U A008311 1716,1287,715,286,78,13,1,1716,3003,2002,1001,364,91,14,1,6435,5005,3003
%N A008311 Triangle of expansions of powers of x in terms of Chebyshev polynomials T_n (x).
%C A008311 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 A008311 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 A008311 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 A008311 Vincenzo Librandi, <a href="/A008311/b008311.txt">Table of n, a(n) for n = 0..5775</a>
%H A008311 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 A008311 H. J. Brothers, <a href="http://www.brotherstechnology.com/docs/Pascal&#39;s_Prism_(supplement).pdf">Pascal's Prism: Supplementary Material</a>.
%H A008311 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A008311 Sum_{k, 0<=k}T(n,k)*cos(kx) = 2^(n-1)*cos(x)^n. - _Philippe Deléham_, Mar 09 2013
%e A008311 Triangle begins:
%e A008311 1;
%e A008311 -, 1;
%e A008311 1, -, 1;
%e A008311 -, 3, -, 1;
%e A008311 3, -, 4, -, 1;
%e A008311 -, 10, -, 5, -, 1;
%e A008311 ...
%e A008311 From _Philippe Deléham_, Mar 09 2013: (Start)
%e A008311 cos(x)      = 1*cos(x),
%e A008311 2*cos(x)^2  = 1 + cos(2x),
%e A008311 4*cos(x)^3  = 3*cos(x) + cos(3x),
%e A008311 8*cos(x)^4  = 3 + 4*cos(2x) + cos(4x),
%e A008311 16*cos(x)^5 = 10*cos(x) + 5*cos(3x) + cos(5x), etc. (End)
%p A008311 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 A008311 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 A008311 Table[row[n], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, May 05 2017, after _R. J. Mathar_ *)
%Y A008311 With zeros: A100257.
%K A008311 nonn,tabf,easy
%O A008311 0,5
%A A008311 _N. J. A. Sloane_
%E A008311 Corrected by _Philippe Deléham_, Nov 12 2005
%E A008311 More terms from _R. J. Mathar_, May 13 2006