A273168 -id:A273168 - cp's OEIS frontend

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

Bradley Klee, May 23 2016

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

			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.

Zak Seidov, No Need For TrigReduce

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

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 *)

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)

A273167 Numerators of coefficient triangle for expansion of x^(2n) in terms of Chebyshev polynomials of the first kind T(2m, x) (A127674).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 5, 15, 3, 1, 35, 7, 7, 1, 1, 63, 105, 15, 45, 5, 1, 231, 99, 495, 55, 33, 3, 1, 429, 3003, 1001, 1001, 91, 91, 7, 1, 6435, 715, 1001, 273, 455, 35, 15, 1, 1, 12155, 21879, 1989, 4641, 1071, 765, 51, 153, 9, 1, 46189, 20995, 62985, 4845, 4845, 969, 4845, 285, 95, 5, 1

Offset: 0

Wolfdieter Lang, Jun 12 2016

The denominator triangle is given in A273168.
The expansion is x^(2*n) = Sum_{m=0..n} R(n, m)*Tnx(2*m, x), n >= 0, with the rational triangle R(n, m) = a(n, m)/A273168(n, m).
  Compare this with A127673.
This is equivalent to the expansion cos(x)^(2n) = Sum_{m=0..n} R(n, m)*cos(2*m*x), n >= 0. Compare this with the even numbered rows of A273496.
See A244420/A244421 for the expansion of x^(2*n+1) in terms of odd-indexed Chebyshev polynomials of the first kind.
The signed rational triangle S(n, m) = R(n, m) * (-1)^m appears in the expansion sin(x)^(2n) = Sum_{m=0..n} S(n, m) * cos(2*m*x), n >= 0. This is equivalent to the identity (1-x^2)^n =  Sum_{m=0..n} S(n, m) * T(2*m, x).

			The triangle a(n, m) begins:
n\m     0     1    2    3    4   5  6   7 8 9
0:      1
1:      1     1
2:      3     1    1
3:      5    15    3    1
4:     35     7    7    1    1
5:     63   105   15   45    5   1
6:    231    99  495   55   33   3  1
7:    429  3003 1001 1001   91  91  7   1
8:   6435   715 1001  273  455  35 15   1 1
9:  12155 21879 1989 4641 1071 765 51 153 9 1
...
The rational triangle R(n, m) begins:
n\m  0       1      2     3      4     5  ...
0:   1
1:  1/2     1/2
2:  3/8     1/2    1/8
3:  5/16   15/32   3/16  1/32
4: 35/128   7/16   7/32  1/16  1/128
5: 63/256 105/256 15/64 45/512 5/256 1/512
...
row 6: 231/1024 99/256 495/2048 55/512 33/1024 3/512 1/2048,
row 7: 429/2048 3003/8192 1001/4096 1001/8192 91/2048 91/8192 7/4096 1/8192,
row 8: 6435/32768 715/2048 1001/4096 273/2048 455/8192 35/2048 15/4096 1/2048 1/32768,
row 9: 12155/65536 21879/65536 1989/8192 4641/32768 1071/16384 765/32768 51/8192 153/131072 9/65536 1/131072,
...
n=3: x^6 = (5/16)*T(0, x) + (15/32)*T(2, x)
  +(3/16)*T(4, x) + (1/32)*T(6,x).
  cos^6(x) = (5/16) + (15/32)*cos(2*x) +
    (3/16)*cos(4*x) + (1/32)*cos(6*x).
  sin^6(x) = (5/16) - (15/32)*cos(2*x) +
    (3/16)*cos(4*x) - (1/32)*cos(6*x).

Index entries for sequences related to Chebyshev polynomials.

Cf. A127673, A127674, A244420, A244421, A273168.

T[MaxN_] := Function[{n}, With[{exp = Expand[(1/2)^(2 n) (Exp[I x] + Exp[-I x])^(2 n)]}, Prepend[ 2 Coefficient[exp, Exp[I 2 # x]] & /@ Range[1, n], exp /. {Exp[_] -> 0}]]][#] & /@ Range[0, MaxN];
T[5] // ColumnForm
T2[MaxN_] := Table[Inverse[Outer[Coefficient[#1, x, #2] &, Prepend[ChebyshevT[#, x] & /@ Range[2 MaxN], 1], Range[0, 2 MaxN]]][[n, m]], {n, 1, 2 MaxN, 2}, {m, 1, n, 2}]
T2[6] // ColumnForm (* Bradley Klee, Jun 14 2016 *)

a(n, m) = if (m == 0, numerator((1/2^(2*n-1)) * binomial(2*n,n)/2), numerator((1/2^(2*n-1))*binomial(2*n, n-m)));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(a(n,k), ", ")); print()); \\ Michel Marcus, Jun 19 2016

a(n, m) = numerator(R(n, m)), n >= 0, m = 1, ..., n, with the rationals R(n, m) given by R(n, 0) = (1/2^(2*n-1))*binomial(2*n,n)/2 and R(n ,m) = (1/2^(2*n-1))*binomial(2*n, n-m) for m =1..n, n >= 0.

A273169 Numerators of coefficient triangle for integrated even powers of cos(x) in terms of x and sin(2mx).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 5, 15, 3, 1, 35, 7, 7, 1, 1, 63, 105, 15, 15, 5, 1, 231, 99, 495, 55, 33, 3, 1, 429, 3003, 1001, 1001, 91, 91, 7, 1, 6435, 715, 1001, 91, 455, 7, 5, 1, 1, 12155, 21879, 1989, 1547, 1071, 153, 17, 153, 9, 1, 46189, 20995, 62985, 1615, 4845, 969, 1615, 285, 95, 5, 1

Offset: 0

Wolfdieter Lang, Jun 13 2016

The denominator triangle is given in A273170.
Int(cos^(2*n)(x), x) = R(n, 0)*x + Sum_{m = 1..n} R(n, m)*sin(2*m*x), n >= 0, with the rational triangle a(n, m)/A273170(n, m).
For the rational triangle for the even powers of cos see A273167/A273168. See also the even-indexed rows of A273496.
For the integral over odd powers of cos see the rational triangle A273171/A273172.
The signed triangle S(n, m) = R(n, m)*(-1)^m appears in the integral of even powers of sin as Int(sin^(2*n)(x), x) = S(n , 0)*x + Sum_{m = 1..n} S(n, m)*sin(2*m*x), n >= 0.

			The triangle a(n, m) begins:
n\m    0     1    2    3    4   5  6   7 8 9
0:     1
1:     1     1
2:     3     1    1
3:     5    15    3    1
4:    35     7    7    1    1
5:    63   105   15   15    5   1
6:   231    99  495   55   33   3  1
7:   429  3003 1001 1001   91  91  7   1
8:  6435   715 1001   91  455   7  5   1 1
9: 12155 21879 1989 1547 1071 153 17 153 9 1
...
row 10: 46189 20995 62985 1615 4845 969 1615 285 95 5 1,
...
The rational triangle R(n, m) begins:
n\m   0      1     2      3      4     ...
0:   1/1
1:   1/2    1/4
2:   3/8    1/4   1/32
3:   5/16  15/64  3/64   1/192
4:  35/128  7/32  7/128  1/96  1/1024
...
row 5: 63/256 105/512 15/256 15/1024 5/2048 1/5120,
row 6: 231/1024 99/512 495/8192 55/3072 33/8192 3/5120 1/24576,
row 7: 429/2048 3003/16384 1001/16384 1001/49152 91/16384 91/81920 7/49152 1/114688,
row 8: 6435/32768 715/4096 1001/16384 91/4096 455/65536 7/4096 5/16384 1/28672 1/524288,
row 9: 12155/65536 21879/131072 1989/32768 1547/65536 1071/131072 153/65536 17/32768 153/1835008 9/1048576 1/2359296,
row 10: 46189/262144 20995/131072 62985/1048576 1615/65536 4845/524288 969/327680 1615/2097152 285/1835008 95/4194304 5/2359296 1/10485760,
...
n = 3: Int(cos^6(x), x) = (5/16)*x + (15/64)*sin(2*x) + (3/64)*sin(4*x) + (1/192)*sin(6*x).
  Int(sin^6(x), x) = (5/16)*x - (15/64)*sin(2*x) + (3/64)*sin(4*x) - (1/192)*sin(6*x).

I. S. Gradstein and I. M. Ryshik, Tables of series, products, and integrals, Volume 1, Verlag Harri Deutsch, 1981, pp. 168-169, 2.513 1. and 3.

Cf. A273170, A273167, A273168, A273496, A273171, A273172.

T[MaxN_] := Function[{n}, With[{exp = Expand[(1/2)^(2 n) (Exp[I x] + Exp[-I x])^(2 n)]}, Prepend[1/# Coefficient[exp, Exp[I 2 # x]] & /@ Range[1, n], exp /. {Exp[_] -> 0}]]][#] & /@ Range[0, MaxN];
T[5] // ColumnForm (* Bradley Klee, Jun 14 2016 *)

a(n, m) = if (m == 0, numerator((1/2^(2*n))*binomial(2*n,n)), numerator((1/2^(2*n))*binomial(2*n, n-m)/m));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(a(n,k), ", ")); print()); \\ Michel Marcus, Jun 19 2016

a(n, m) = numerator(R(n, m)) with the rationals R(n, m) defined by R(n, 0) = (1/2^(2*n))*binomial(2*n,n) and R(n, m) = (1/2^(2*n))*binomial(2*n, n-m)/m for m = 1, ..., n, n >= 0. See the Gradstein-Ryshik reference (where the sin arguments are falling).

cp's OEIS Frontend

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).

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A273167 Numerators of coefficient triangle for expansion of x^(2n) in terms of Chebyshev polynomials of the first kind T(2m, x) (A127674).

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A273169 Numerators of coefficient triangle for integrated even powers of cos(x) in terms of x and sin(2mx).

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cp's OEIS Frontend

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).

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Examples

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Crossrefs

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Mathematica

Formula

A273167 Numerators of coefficient triangle for expansion of x^(2*n) in terms of Chebyshev polynomials of the first kind T(2*m, x) (A127674).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A273169 Numerators of coefficient triangle for integrated even powers of cos(x) in terms of x and sin(2*m*x).

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

PARI

Formula

A273167 Numerators of coefficient triangle for expansion of x^(2n) in terms of Chebyshev polynomials of the first kind T(2m, x) (A127674).

A273169 Numerators of coefficient triangle for integrated even powers of cos(x) in terms of x and sin(2mx).