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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).

%I A028298 #32 Jul 15 2024 07:28:51
%S A028298 1,2,3,-4,4,-8,5,-20,16,6,-32,32,7,-56,112,-64,8,-80,192,-128,9,-120,
%T A028298 432,-576,256,10,-160,672,-1024,512,11,-220,1232,-2816,2816,-1024,12,
%U A028298 -280,1792,-4608,5120,-2048,13,-364,2912,-9984,16640,-13312,4096,14,-448,4032,-15360,28160,-24576,8192,15,-560,6048
%N 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).
%C A028298 Rows have ceiling(n/2) terms.
%D A028298 I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th ed., Section 1.335, p. 35.
%F A028298 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
%e A028298 Triangle begins:
%e A028298    1;
%e A028298    2;
%e A028298    3,   -4;
%e A028298    4,   -8;
%e A028298    5,  -20,  16;
%e A028298    6,  -32,  32;
%e A028298    7,  -56, 112,   -64;
%e A028298    8,  -80, 192,  -128;
%e A028298    9, -120, 432,  -576, 256;
%e A028298   10, -160, 672, -1024, 512;
%e A028298   ...
%e A028298 sin 3x = 3 sin x - 4 sin^3 x;
%e A028298 sin 4x / cos x = 4 sin x - 8 sin^3 x, etc.
%t A028298 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 *)
%o A028298 (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 */
%Y A028298 Cf. A028297.
%K A028298 sign,tabf,nice,easy
%O A028298 1,2
%A A028298 _N. J. A. Sloane_
%E A028298 More terms from Larry Reeves (larryr(AT)acm.org), Sep 08 2000