A028298 Triangle of coefficients in expansion of sin(n*x) (or sin(n*x)/cos(x) if n is even) in ascending powers of sin(x).

1, 2, 3, -4, 4, -8, 5, -20, 16, 6, -32, 32, 7, -56, 112, -64, 8, -80, 192, -128, 9, -120, 432, -576, 256, 10, -160, 672, -1024, 512, 11, -220, 1232, -2816, 2816, -1024, 12, -280, 1792, -4608, 5120, -2048, 13, -364, 2912, -9984, 16640, -13312, 4096, 14, -448, 4032, -15360, 28160, -24576, 8192, 15, -560, 6048

Offset

1,2

Comments

Rows have ceiling(n/2) terms.

References

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th ed., Section 1.335, p. 35.

Links

Table of n, a(n) for n=1..59.

Formula

T(n,k) = (-1)^k*((n-2*k)*(-1)^n-n-2*k)/(2*n+(-1)^n-1+4*k)*2^(2*k+((-1)^n-1)/2)*binomial((2*n+(-1)^n-1)/4+k,(2*n-(-1)^n+1)/4-k). - Tani Akinari, Jul 15 2024

Example

Triangle begins:
   1;
   2;
   3,   -4;
   4,   -8;
   5,  -20,  16;
   6,  -32,  32;
   7,  -56, 112,   -64;
   8,  -80, 192,  -128;
   9, -120, 432,  -576, 256;
  10, -160, 672, -1024, 512;
  ...
sin 3x = 3 sin x - 4 sin^3 x;
sin 4x / cos x = 4 sin x - 8 sin^3 x, etc.

Mathematica

t[n_] := (Sin[n x]/If[EvenQ[n], Cos[x], 1] // TrigExpand) /. Cos[x]^m_ /; EvenQ[m] -> (1 - Sin[x]^2)^(m/2) // Expand; Flatten[Table[ Partition[ CoefficientList[t[n], Sin[x]] , 2][[All, 2]], {n, 1, 15}]][[1 ;; 59]]  (* Jean-François Alcover, May 06 2011 *)

Prog

(Maxima) T(n, k):=(-1)^k*((n-2*k)*(-1)^n-n-2*k)/(2*n+(-1)^n-1+4*k)*2^(2*k+((-1)^n-1)/2)*binomial((2*n+(-1)^n-1)/4+k, (2*n-(-1)^n+1)/4-k); /* Tani Akinari, Jul 15 2024 */

Crossrefs

Cf. A028297.

Sequence in context: A301763 A236129 A240219 * A047966 A360243 A360683

Adjacent sequences: A028295 A028296 A028297 * A028299 A028300 A028301

Keyword

sign,tabf,nice,easy

Author

N. J. A. Sloane

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Sep 08 2000

Status

approved