A244421 -id:A244421 - 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)

A244420 Numerators of coefficient triangle for expansion of x^n in terms of polynomials Todd(k, x) = T(2*k+1, sqrt(x))/sqrt(x) (A084930), with the Chebyshev polynomials of the first kind (type T).

Original entry on oeis.org

1, 3, 1, 5, 5, 1, 35, 21, 7, 1, 63, 21, 9, 9, 1, 231, 165, 165, 55, 11, 1, 429, 1287, 715, 143, 39, 13, 1, 6435, 5005, 3003, 1365, 455, 105, 15, 1, 12155, 2431, 1547, 1547, 595, 85, 17, 17, 1, 46189, 37791, 12597, 6783, 2907, 969, 969, 171, 19, 1, 88179, 146965, 101745, 14535, 6783, 20349, 5985, 665, 105, 21, 1

Offset: 0

Wolfdieter Lang, Aug 04 2014

This expansion is due to the Riordan property of the triangle A084930. The inverse of the lower triangular matrix built by A084930 is therefore also a (rational) Riordan triangle, namely ((2 - c(z/4))/(1-z), -1 + c(z/4)) in the standard notation, where c is the o.g.f. of A000108 (Catalan).
For the denominators of this triangle see A244421.
The expansion is x^n = sum(R(n,m)*Todd(m, x), m=0..n), n >= 0, with the rational triangle with entries R(n,m) = a(n, m)/b(n, m) with b(n, m) = A244421(n, m).
If one uses instead the expansion of (4*x)^n one finds the integer triangle A111418: (4*x)^n = sum(A111418(n,k) * Todd(k, x), k=0..n).
The row sums of the rational triangle R(n,m) are identically 1. The alternating row sums have o.g.f. 1/sqrt(1-x) which generates A001790(n)/A046161(n) (see a Michael Somos comment on A046161), namely 1, 1/2, 3/8, 5/16, 35/128, 63/256, 231/1024, 429/2048, ...
From Wolfdieter Lang, Jun 13 2016: (Start)
R(n,m) = a(n, m)/ A244421(n, m) is also the rational triangle for the expansion cos^{2*n+1}(x) = Sum_{m=0..n} R(n, m)*cos((2*m+1)*x), n >= 0, m = 0..n. Compare with the odd numbered rows of A273496. In terms of Chebyshev T-polynomials (A053120) this is the identity x^(2*n+1) = Sum_{m=0..n} R(n, m)*T(2*m+1, x).
S(n,m) = (-1)^m*a(n, m)/ A244421(n, m) is the rational triangle for the expansion sin^{2*n+1}(x) = Sum_{m=0..n} S(n, m)*sin((2*m+1)*x), n >= 0, m = 0..n. In terms of Chebyshev S-polynomials (A049310) this is equivalent to the identity (4 - x^2)*n = Sum_{m=0..n} (-1)^m * binomial(n, n-m)*S(2*m,x), n >= 0.
(End)

			The numerator triangle a(n,m) begins:
  n\m      0      1      2     3    4     5    6   7   8   9
  0:       1
  1:       3      1
  2:       5      5      1
  3:      35     21      7     1
  4:      63     21      9     9    1
  5:     231    165    165    55   11     1
  6:     429   1287    715   143   39    13    1
  7:    6435   5005   3003  1365  455   105   15   1
  8:   12155   2431   1547  1547  595    85   17  17   1
  9:   46189  37791  12597  6783 2907   969  969 171  19   1
  ...
The rational triangle R(n,m) begins:
  n\m       0        1         2       3        4        5
  0:        1
  1:      3/4      1/4
  2:      5/8     5/16      1/16
  3:    35/64    21/64      7/64    1/64
  4:   63/128    21/64      9/64   9/256    1/256
  5:  231/512  165/512  165/1024 55/1024  11/1024   1/1024
  ...
The next rows are:
  n=6: 429/1024, 1287/4096, 715/4096, 143/2048, 39/2048, 13/4096, 1/4096,
  n=7: 6435/16384, 5005/16384, 3003/16384, 1365/16384, 455/16384, 105/16384, 15/16384, 1/16384,
  n=8: 12155/32768, 2431/8192, 1547/8192, 1547/16384, 595/16384, 85/8192, 17/8192, 17/65536, 1/65536,
  n=9: 46189/131072, 37791/131072, 12597/65536, 6783/65536, 2907/65536, 969/65536, 969/262144, 171/262144, 19/262144, 1/262144,
  ...
Expansions:
x^2 = 5/8 * Todd(0,x) + 5/16 * Todd(1,x) + 1/16 * Todd(2,x) = 5/8 + (5/16)*(-3 + 4*x) +(1/16)*(5 -20*x + 16*x^2).
x^3 = (35*Todd(0, x) + 21*Todd(1, x) + 7*Todd(2, x) + 1*Todd(3, x))/64 = (35 + 21*(-3+4*x) + 7*( 5-20*x+16*x^2) + (-7+56*x-112*x^2+64*x^3))/64.
For the Todd polynomials see the coefficient table A084930.

Wolfdieter Lang, First rows of the triangles.

Cf. A084930, A244421, A000108, A001790, A046161, A111418, A273496.

a(n, m) = numerator(R(n, m)) with the rationals Riordan matrix elements R(n, m)= [x^m]R(n, x), with the row polynomials R(n, x) generated by ((2 - c(z/4))/(1-z))/(1 - x*(-1 + c(z/4))) = 2*((1+x)*(z-1) + (1-x)*sqrt(1-z))/((1-z)*((1+x)^2*z - 4*x)), where c(x) is the o.g.f. of the Catalan numbers A000108.

The rationals R(n, m) = binomial(2*n+1, m)/2^(2*n). - Wolfdieter Lang, Jun 12 2016

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.

A273171 Triangle for numerators of coefficients for integrated odd powers of cos(x) in terms sin((2m+1)x).

Original entry on oeis.org

1, 3, 1, 5, 5, 1, 35, 7, 7, 1, 63, 7, 9, 9, 1, 231, 55, 33, 55, 11, 1, 429, 429, 143, 143, 13, 13, 1, 6435, 5005, 3003, 195, 455, 105, 15, 1, 12155, 2431, 1547, 221, 595, 85, 17, 17, 1, 46189, 12597, 12597, 969, 323, 969, 969, 57, 19, 1, 88179, 146965, 20349, 14535, 2261, 20349, 5985, 133, 105, 21, 1

Offset: 0

Wolfdieter Lang, Jun 13 2016

The triangle for the denominators is given in A273172.
Int(cos^(2*n+1)(x), x) = Sum_{m = 0..n} R(n, m)*sin((2*m+1)*x), n >= 0, with the rational triangle a(n, m)/A273172(n, m).
For the rational triangle for the odd powers of cos see A244420/A244421. See also the odd-indexed rows of A273496.
The signed rational triangle S(n, m) = R(n, m)*(-1)^(m+1) appears in the formula
  Int(sin^(2*n+1)(x), x) = Sum_{m = 0..n} S(n, m)*cos((2*m+1)*x), n >= 0,

			The triangle a(n, m) begins:
n\m    0     1     2   3   4   5   6  7  8 9
0:     1
1:     3     1
2:     5     5     1
3:    35     7     7   1
4:    63     7     9   9  1
5:   231    55    33  55  11   1
6:   429   429   143 143  13  13   1
7:  6435  5005  3003 195 455 105  15  1
8: 12155  2431  1547 221 595  85  17 17  1
9: 46189 12597 12597 969 323 969 969 57 19 1
...
row 10: 88179 146965 20349 14535 2261 20349 5985 133 105 21 1,
...
The rational triangle R(n, m) begins:
n\m   0    1     2    3      4     ...
0:   1/1
1:   3/4  1/12
2:   5/8  5/48 1/80
3:  35/64 7/64 7/320 1/448
4: 63/128 7/64 9/320 9/1792 1/2304
...
row 5: 231/512 55/512 33/1024 55/7168 11/9216 1/11264,
row 6: 429/1024 429/4096 143/4096 143/14336 13/6144 13/45056 1/53248,
row 7: 6435/16384 5005/49152 3003/81920 195/16384 455/147456 105/180224 15/212992 1/245760,
row 8: 12155/32768 2431/24576 1547/40960 221/16384 595/147456 85/90112 17/106496 17/983040 1/1114112,
row 9: 46189/131072 12597/131072 12597/327680 969/65536 323/65536 969/720896 969/3407872 57/1310720 19/4456448 1/4980736,
row 10: 88179/262144 146965/1572864 20349/524288 14535/917504 2261/393216 20349/11534336 5985/13631488 133/1572864 105/8912896 21/19922944 1/22020096.
...
n = 3: Int(cos^7(x), x) = (35/64)*sin(x) + (7/64)*sin(3*x) + (7/320)*sin(5*x) + (1/448)*sin(7*x). Gradstein-Rhyshik, p. 169, 2.513 16.
  Int(sin^7(x), x) = -(35/64)*cos(x) + (7/64)*cos(3*x) - (7/320)*cos(5*x) + (1/448)*cos(7*x). Gradstein-Rhyshik, p. 169, 2.513 10.

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 4.

Cf. A273172, A244420, A244421, A273496.

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

a(n, m) = numerator((1/2^(2*n))*binomial(2*n+1, n-m)/(2*m+1));
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) = (1/2^(2*n)) * binomial(2*n+1, n-m)/(2*m+1) for m = 0, ..., 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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A244420 Numerators of coefficient triangle for expansion of x^n in terms of polynomials Todd(k, x) = T(2*k+1, sqrt(x))/sqrt(x) (A084930), with the Chebyshev polynomials of the first kind (type T).

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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).

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A273171 Triangle for numerators of coefficients for integrated odd powers of cos(x) in terms sin((2*m+1)*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).

A273171 Triangle for numerators of coefficients for integrated odd powers of cos(x) in terms sin((2m+1)x).