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

0,2

Comments

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

Links

Table of n, a(n) for n=0..65.

Formula

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).

Example

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.

Prog

(PARI) 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

Crossrefs

Cf. A273171.

Sequence in context: A156681 A231100 A229179 * A307853 A334768 A247327

Adjacent sequences: A273169 A273170 A273171 * A273173 A273174 A273175

Keyword

nonn,tabl,frac,easy

Author

Wolfdieter Lang, Jun 13 2016

Status

approved