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 A008311 Triangle of expansions of powers of x in terms of Chebyshev polynomials T_n (x). 5
 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, 9, 1, 126, 210, 120, 45, 10, 1, 462, 330, 165, 55, 11, 1, 462, 792, 495, 220, 66, 12, 1, 1716, 1287, 715, 286, 78, 13, 1, 1716, 3003, 2002, 1001, 364, 91, 14, 1, 6435, 5005, 3003 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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 For n>=2 found in A002378, a(n)=A034869(n)/2, for all others a(n)=A034869(n). - R. J. Mathar, May 13 2006 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. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..5775 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]. H. J. Brothers, Pascal's Prism: Supplementary Material. FORMULA Sum_{k, 0<=k}T(n,k)*cos(kx) = 2^(n-1)*cos(x)^n. - Philippe Deléham, Mar 09 2013 EXAMPLE Triangle begins: 1; -, 1; 1, -, 1; -, 3, -, 1; 3, -, 4, -, 1; -, 10, -, 5, -, 1; ... From Philippe Deléham, Mar 09 2013: (Start) cos(x)      = 1*cos(x), 2*cos(x)^2  = 1 + cos(2x), 4*cos(x)^3  = 3*cos(x) + cos(3x), 8*cos(x)^4  = 3 + 4*cos(2x) + cos(4x), 16*cos(x)^5 = 10*cos(x) + 5*cos(3x) + cos(5x), etc. (End) MAPLE 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 MATHEMATICA 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}]]; Table[row[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, May 05 2017, after R. J. Mathar *) CROSSREFS With zeros: A100257. Sequence in context: A093560 A173934 A131504 * A175721 A337936 A296388 Adjacent sequences:  A008308 A008309 A008310 * A008312 A008313 A008314 KEYWORD nonn,tabf,easy AUTHOR EXTENSIONS Corrected by Philippe Deléham, Nov 12 2005 More terms from R. J. Mathar, May 13 2006 STATUS approved

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