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%I #12 Jan 31 2017 10:59:00
%S 1,2,0,3,0,-1,4,0,-4,0,5,0,-10,0,1,6,0,-20,0,6,0,7,0,-35,0,21,0,-1,8,
%T 0,-56,0,56,0,-8,0,9,0,-84,0,126,0,-36,0,1,10,0,-120,0,252,0,-120,0,
%U 10,0,11,0,-165,0,462,0,-330,0,55,0,-1,12,0,-220,0,792,0,-792,0,220,0,-12,0,13,0,-286,0,1287,0
%N Triangle read by rows giving coefficients of the trigonometric expansion of sin(n*x).
%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Kimberling/kimberling56.html">Polynomials associated with reciprocation</a>, JIS 12 (2009) 09.3.4, section 5.
%F T(n,k) = C(n+1,k+1)*sin(Pi*(k+1)/2). - _Paul Barry_, May 21 2006
%e The trigonometric expansion of sin(4x) is 4*cos(x)^3*sin(x) - 4*cos(x)*sin(x)^3, so the fourth row is 4, 0, -4, 0.
%e Triangle begins:
%e 1
%e 2 0
%e 3 0 -1
%e 4 0 -4 0
%e 5 0 -10 0 1
%e 6 0 -20 0 6 0
%e 7 0 -35 0 21 0 -1
%e 8 0 -56 0 56 0 -8 0
%t Flatten[ Table[ Plus @@ CoefficientList[ TrigExpand[ Sin[n*x]], {Sin[x], Cos[x]}], {n, 13}]]
%Y First column is A000027 = C(n, 1), third column is A000292 = C(n, 3), fifth column is A000389 = C(n, 5), seventh column is A000580 = C(n, 7), ninth column is A000582 = C(n, 9).
%Y A001288 = C(n, 11), A010966 = C(n, 13), A010968 = C(n, 15), A010970 = C(n, 17), A010972 = C(n, 19),
%Y A010974 = C(n, 21), A010976 = C(n, 23), A010978 = C(n, 25), A010980 = C(n, 27), A010982 = C(n, 29),
%Y A010984 = C(n, 31), A010986 = C(n, 33), A010988 = C(n, 35), A010990 = C(n, 37), A010992 = C(n, 39),
%Y A010994 = C(n, 41), A010996 = C(n, 43), A010998 = C(n, 45), A011000 = C(n, 47), A017713 = C(n, 49)
%Y Another version of the triangle in A034867. Cf. A096754.
%Y A017715 = C(n, 51), A017717 = C(n, 53), A017719 = C(n, 55), A017721 = C(n, 57), etc.
%K sign,tabl
%O 1,2
%A _Robert G. Wilson v_, Jul 06 2004
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