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

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

Author

Wolfdieter Lang, Jun 13 2016

Keywords

• nonn
• tabl
• frac
• easy

Comments

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,

Examples

			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.
		

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 4.

Crossrefs

Cf. A273172, A244420, A244421, A273496.

Programs

Mathematica

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 *)

PARI

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

Formula

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).