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%I #26 Mar 15 2020 03:33:16
%S 1,1,0,1,0,1,1,0,3,0,1,0,4,0,3,1,0,5,0,10,0,1,0,6,0,15,0,10,1,0,7,0,
%T 21,0,35,0,1,0,8,0,28,0,56,0,35,1,0,9,0,36,0,84,0,126,0,1,0,10,0,45,0,
%U 120,0,210,0,126,1,0,11,0,55,0,165,0,330,0,462,0,1,0,12,0,66,0,220,0
%N Triangle of expansions of 2^(k-1)*x^k in terms of T(n,x), in descending degrees n of T, with T the Chebyshev polynomials.
%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="/A100257/b100257.txt">Table of n, a(n) for n = 0..6104</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/math/pascals-prism.html">Pascal's Prism: Supplementary Material</a>.
%H Daniel J. Greenhoe, <a href="https://www.researchgate.net/publication/337858762_Frames_and_Bases_Structure_and_Design_version_020">Frames and Bases: Structure and Design</a>, Version 0.20, Signal Processing ABCs series (2019) Vol. 4, see page 175.
%H Daniel J. Greenhoe, <a href="https://www.researchgate.net/publication/337858659_A_Book_Concerning_Transforms_version_010">A Book Concerning Transforms</a>, Version 0.10, Signal Processing ABCs series (2019) Vol. 5, see page 97.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%e x^0 = T(0,x)
%e x^1 = T(1,x) + 0T(0,x)
%e 2x^2 = T(2,x) + 0T(1,x) + 1T(0,x)
%e 4x^3 = T(3,x) + 0T(2,x) + 3T(1,x) + 0T(0,x)
%e 8x^4 = T(4,x) + 0T(3,x) + 4T(2,x) + 0T(1,x) + 3T(0,x)
%e 16x^5 = T(5,x) + 0T(4,x) + 5T(3,x) + 0T(2,x) + 10T(1,x) + 0T(0,x)
%t a[k_, n_] := If[k == 1, 1, If[EvenQ[n] || k < 0 || n > k, 0, If[n >= k - 1, Binomial[2*Floor[k/2], Floor[k/2]]/2, Binomial[k - 1, Floor[n/2]]]]];
%t Table[a[k, n], {k, 1, 13}, {n, 1, k}] // Flatten (* _Jean-François Alcover_, May 04 2017, translated from PARI *)
%o (PARI) a(k,n)=if(k==1,1,if(n%2==0||k<0||n>k,0,if(n>=k-1,binomial(2*floor(k/2),floor(k/2))/2,binomial(k-1,floor(n/2)))))
%Y Without zeros: A008311. Row sums are A011782. Cf. A092392.
%Y Diagonals are (with interleaved zeros) twice A001700, A001791, A002054, A002694, A003516, A002696, A030053, A004310, A030054, A004311, A030055, A004312, A030056, A004313.
%K nonn,tabl
%O 0,9
%A _Ralf Stephan_, Nov 13 2004
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