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URL: https://oeis.org/A273496

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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).
16
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
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 *)
KEYWORD
nonn,tabl
AUTHOR
Bradley Klee, May 23 2016
STATUS
approved