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

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

Author

Wolfdieter Lang, Jun 13 2016

Keywords

• nonn
• tabl
• frac
• easy

Comments

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.

Examples

			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).
		

References

• 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.

Crossrefs

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

Programs

Mathematica

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 *)

PARI

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

Formula

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).