A273496 Triangle read by rows: coefficients in the expansion cos(x)^n = (1/2)^n * Sum_{k=0..n} T(n,k) * cos(k*x).

1, 0, 2, 2, 0, 2, 0, 6, 0, 2, 6, 0, 8, 0, 2, 0, 20, 0, 10, 0, 2, 20, 0, 30, 0, 12, 0, 2, 0, 70, 0, 42, 0, 14, 0, 2, 70, 0, 112, 0, 56, 0, 16, 0, 2, 0, 252, 0, 168, 0, 72, 0, 18, 0, 2, 252, 0, 420, 0, 240, 0, 90, 0, 20, 0, 2

Offset

0,3

Comments

These coefficients are especially useful when integrating powers of cosine x (see examples).

Nonzero, even elements of the first column are given by A000984; T(2n,0) = binomial(2n,n).

For the rational triangles for even and odd powers of cos(x) see A273167/A273168 and A244420/A244421, respectively. - Wolfdieter Lang, Jun 13 2016

Mathematica needs no TrigReduce to integrate Cos[x]^k. See link. - Zak Seidov, Jun 13 2016

Links

Table of n, a(n) for n=0..65.

Zak Seidov, No Need For TrigReduce

Formula

From Robert Israel, May 24 2016: (Start)

T(n,k) = 0 if n-k is odd.

T(n,0) = binomial(n,n/2) if n is even.

T(n,k) = 2*binomial(n,(n-k)/2) otherwise. (End)

Example

n/k|  0   1   2   3   4   5   6
-------------------------------
0  |  1
1  |  0   2
2  |  2   0   2
3  |  0   6   0   2
4  |  6   0   8   0   2
5  |  0   20  0   10  0   2
6  |  20  0   30  0   12  0   2
-------------------------------
cos(x)^4 = (1/2)^4 (6 + 8 cos(2x) + 2 cos(4x)).
I4 = Int dx cos(x)^4 = (1/2)^4 Int dx ( 6 + 8 cos(2x) + 2 cos(4x) ) = C + 3/8 x + 1/4 sin(2x) + 1/32 sin(4x).
Over range [0,2Pi], I4 = (3/4) Pi.

Mathematica

T[MaxN_] := Function[{n}, With[
       {exp = Expand[Times[ 2^n, TrigReduce[Cos[x]^n]]]},
       Prepend[Coefficient[exp, Cos[# x]] & /@ Range[1, n],
        exp /. {Cos[_] -> 0}]]][#] & /@ Range[0, MaxN]; Flatten@T[10]
(* alternate program *)
T2[MaxN_] := Function[{n}, With[{exp = Expand[(Exp[I x] + Exp[-I x])^n]}, Prepend[2 Coefficient[exp, Exp[I # x]] & /@ Range[1, n], exp /. {Exp[_] -> 0}]]][#] & /@ Range[0, MaxN]; T2[10] // ColumnForm (* Bradley Klee, Jun 13 2016 *)

Crossrefs

Cf. A000984, A001790, A046161, A038533, A038534, A273506, A273507, A273167, A273168, A244420, A244421.

Sequence in context: A371711 A193863 A363566 * A286576 A322523 A333210

Adjacent sequences: A273493 A273494 A273495 * A273497 A273498 A273499

Keyword

nonn,tabl

Author

Bradley Klee, May 23 2016

Status

approved