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

Author

N. J. A. Sloane

Keywords

• nonn
• tabf
• easy

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

Examples

			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)
		

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.

Crossrefs

With zeros: A100257.

Programs

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 *)

Formula

Sum_{k, 0<=k}T(n,k)*cos(kx) = 2^(n-1)*cos(x)^n. - Philippe Deléham, Mar 09 2013

Extensions

Corrected by Philippe Deléham, Nov 12 2005

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