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

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

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.

Index entries for sequences related to Chebyshev polynomials.

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

N. J. A. Sloane

Extensions

Corrected by Philippe Deléham, Nov 12 2005

More terms from R. J. Mathar, May 13 2006

Status

approved