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%I #6 Jul 14 2016 13:18:10
%S 1,1,0,-1,1,0,-3,1,0,-6,0,1,1,0,-10,0,5,1,0,-15,0,15,0,-1,1,0,-21,0,
%T 35,0,-7,1,0,-28,0,70,0,-28,0,1,1,0,-36,0,126,0,-84,0,9,1,0,-45,0,210,
%U 0,-210,0,45,0,-1,1,0,-55,0,330,0,-462,0,165,0,-11,1,0,-66,0,495,0,-924,0,495,0,-66,0,1,1,0,-78,0,715
%N Triangle read by rows giving coefficients of the trigonometric expansion of Cos(n*x).
%C T(n,k)=cos(n,k)*cos(pi*k/2) begins {1}, {1,0}, {1,0,-1}, {1,0,-3,0},... - _Paul Barry_, May 21 2006
%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.
%e The trigonometric expansion of Cos(4x) = Cos[x]^4 - 6*Cos[x]^2*Sin[x]^2 + Sin[x]^4, therefore the fourth row is 1, 0, -6, 0, 1.
%e The trigonometric expansion of Cos(5x) = Cos[x]^5 - 10*Cos[x]^3*Sin[x]^2 + 5*Cos[x]*Sin[x]^4, therefore the fifth row of the triangle is 1, 0, -10, 0, 5
%e The table begins:
%e 1
%e 1 0 -1
%e 1 0 -3
%e 1 0 -6 0 1
%e 1 0 -10 0 5
%e 1 0 -15 0 15 0 -1
%e 1 0 -21 0 35 0 -7
%e 1 0 -28 0 70 0 -28 0 1
%t Flatten[Table[ Plus @@ CoefficientList[ TrigExpand[ Cos[n*x]], { Cos[x], Sin[x]}], {n, 13}]]
%Y Another version of the triangle in A034839. Cf. A095704.
%Y First column is A000012 = C(n, 0), third column is A000217 = C(n, 2), fifth column is A000332 = C(n, 4), seventh column is A000579 = C(n, 6), ninth column is A000581 = C(n, 8).
%Y A001287 = C(n, 10), A010965 = C(n, 12), A010967 = C(n, 14), A010969 = C(n, 16), A010971 = C(n, 18),
%Y A010973 = C(n, 20), A010975 = C(n, 22), A010977 = C(n, 24), A010979 = C(n, 26), A010981 = C(n, 28),
%Y A010983 = C(n, 30), A010985 = C(n, 32), A010987 = C(n, 34), A010989 = C(n, 36), A010991 = C(n, 38),
%Y A010993 = C(n, 40), A010995 = C(n, 42), A010997 = C(n, 44), A010999 = C(n, 46), A011001 = C(n, 48),
%Y A017714 = C(n, 50), A017716 = C(n, 52), A017718 = C(n, 54), A017720 = C(n, 56), etc.
%K sign,tabl
%O 1,7
%A _Robert G. Wilson v_, Jul 07 2004
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