A273170 Denominators of coefficient triangle for integrated even powers of cos(x) in terms of x and sin(2*m*x).

1, 2, 4, 8, 4, 32, 16, 64, 64, 192, 128, 32, 128, 96, 1024, 256, 512, 256, 1024, 2048, 5120, 1024, 512, 8192, 3072, 8192, 5120, 24576, 2048, 16384, 16384, 49152, 16384, 81920, 49152, 114688, 32768, 4096, 16384, 4096, 65536, 4096, 16384, 28672, 524288, 65536, 131072, 32768, 65536, 131072, 65536, 32768, 1835008, 1048576, 2359296, 262144, 131072, 1048576, 65536, 524288, 327680, 2097152, 1835008, 4194304, 2359296, 10485760

Offset: 0

Author

Wolfdieter Lang, Jun 13 2016

Keywords

• nonn
• tabl
• frac
• easy

Comments

See the numerator triangle A273169, also for the formula of int(cos^(2*n)(x), x) in terms of x and sin(2*m*x).

Examples

			See A273169, also for the rationals R(n,m).
The triangle a(n, m) begins:
n\m    0   1    2    3    4    5     6 ...
0:     1
1:     2   4
2:     8   4   32
3:    16  64   64  192
4:   128  32  128   96 1024
5:   256 512  256 1024 2048 5120
6:  1024 512 8192 3072 8192 5120 24576
...
row 7: 2048 16384 16384 49152 16384 81920 49152 114688,
row 8: 32768 4096 16384 4096 65536 4096 16384 28672 524288,
row 9: 65536 131072 32768 65536 131072 65536 32768 1835008 1048576 2359296,
row 10: 262144 131072 1048576 65536 524288 327680 2097152 1835008 4194304 2359296 10485760,
...
		

Crossrefs

Cf. A273169.

Programs

PARI

a(n, m) = if (m == 0, denominator((1/2^(2*n))*binomial(2*n,n)), denominator((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) = denominator(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 given in A273169 (where the sin arguments are falling).