A273171 -id:A273171 - cp's OEIS frontend

A273172 Triangle for denominators of coefficients for integrated odd powers of cos(x) in terms sin((2m+1)x).

1, 4, 12, 8, 48, 80, 64, 64, 320, 448, 128, 64, 320, 1792, 2304, 512, 512, 1024, 7168, 9216, 11264, 1024, 4096, 4096, 14336, 6144, 45056, 53248, 16384, 49152, 81920, 16384, 147456, 180224, 212992, 245760, 32768, 24576, 40960, 16384, 147456, 90112, 106496, 983040, 1114112, 131072, 131072, 327680, 65536, 65536, 720896, 3407872, 1310720, 4456448, 4980736, 262144, 1572864, 524288, 917504, 393216, 11534336, 13631488, 1572864, 8912896, 19922944, 22020096

Offset: 0

Wolfdieter Lang, Jun 13 2016

For the numerator triangle see A273171, also for the cos^(2*n+1) formula, and the Gradstein-Ryshik reference.

			The triangle a(n, m) begins:
n\m    0    1    2     3    4     5     6 ...
0:     1
1:     4   12
2:     8   48   80
3:    64   64  320   448
4:   128   64  320  1792 2304
5:   512  512 1024  7168 9216 11264
6:  1024 4096 4096 14336 6144 45056 53248
...
row 7: 16384 49152 81920 16384 147456 180224 212992 245760,
row 8: 32768 24576 40960 16384 147456 90112 106496 983040 1114112,
row 9: 131072 131072 327680 65536 65536 720896 3407872 1310720 4456448 4980736,
row 10: 262144 1572864 524288 917504 393216 11534336 13631488 1572864 8912896 19922944 22020096,...
For the head of the rational triangle see A273171.

Cf. A273171.

a(n, m) = denominator((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) = denominator(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 given in A273171 (where the sin arguments are falling).

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

A273172 Triangle for denominators of coefficients for integrated odd powers of cos(x) in terms sin((2m+1)x).

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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

A273172 Triangle for denominators of coefficients for integrated odd powers of cos(x) in terms sin((2*m+1)*x).

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PARI

Formula

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

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Author

Keywords

Comments

Examples

References

Crossrefs

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Mathematica

PARI

Formula

A273172 Triangle for denominators of coefficients for integrated odd powers of cos(x) in terms sin((2m+1)x).

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