editing

approved

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.

approved

editing

T(n,k)=cos(n,k)*cos(pi*k/2) begins {1}, {1,0}, {1,0,-1}, {1,0,-3,0},... - _Paul Barry (pbarry(AT)wit.ie), _, May 21 2006

_Robert G. Wilson v (rgwv(AT)rgwv.com), _, Jul 07 2004

T(n,k)=cos(n,k)*cos(pi*k/2) begins {1}, {1,0}, {1,0,-1}, {1,0,-3,0},... - Paul Barry (pbarry(AT)wit.ie), May 21 2006

sign,tabl,new

Triangle read by rows giving coefficients of the trigonometric expansion of Cos(n*x).

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, 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, 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

1,7

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.

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

The table begins:

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 35 0 -7

1 0 -28 0 70 0 -28 0 1

Flatten[Table[ Plus @@ CoefficientList[ TrigExpand[ Cos[n*x]], { Cos[x], Sin[x]}], {n, 13}]]

Another version of the triangle in A034839. Cf. A095704.

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).

A001287 = C(n, 10), A010965 = C(n, 12), A010967 = C(n, 14), A010969 = C(n, 16), A010971 = C(n, 18),

A010973 = C(n, 20), A010975 = C(n, 22), A010977 = C(n, 24), A010979 = C(n, 26), A010981 = C(n, 28),

A010983 = C(n, 30), A010985 = C(n, 32), A010987 = C(n, 34), A010989 = C(n, 36), A010991 = C(n, 38),

A010993 = C(n, 40), A010995 = C(n, 42), A010997 = C(n, 44), A010999 = C(n, 46), A011001 = C(n, 48),

A017714 = C(n, 50), A017716 = C(n, 52), A017718 = C(n, 54), A017720 = C(n, 56), etc.

sign,tabl

Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 07 2004

approved