A164658(n,m)/A164659(n,m) =: r(n,m) W. Lang, Oct 16, 2009 Triangle of signed rational coefficients r(n,m) of the indefinite integral (antiderivative) of Chebyshev's polynomials of the first kind: int(T(n,x),x) = sum(r(n,m)*x^m,m=1..n+1), n>=0. Numerator triangle A164658(n,m): n\m 1 2 3 4 5 6 7 8 9 10 ... 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 2 -1 0 2 0 0 0 0 0 0 0 3 0 -3 0 1 0 0 0 0 0 0 4 1 0 -8 0 8 0 0 0 0 0 5 0 5 0 -5 0 8 0 0 0 0 6 -1 0 6 0 -48 0 32 0 0 0 7 0 -7 0 14 0 -56 0 8 0 0 8 1 0 -32 0 32 0 -256 0 128 0 9 0 9 0 -30 0 72 0 -72 0 128 . . . Row sums give A164662: [1, 1, 1, -2, 1, 8, -11, -41, -127, 107, -639, -1372, -3695, 514, -25983, -26339, -70655, -46299, -430955, -484134, -2808479, 93148, -5032895, -17319181, -72165695, 43371103, -171203135, -378398576, -148383647, -2605023034, -3368133419, 11479942073, -11902375935, 2021161097, -708801692671, -286168167562, 1273953448369, 1048426172704, 613803430529, -12665985704225, ...]. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Denominator triangle A164659(n,m) a(n,m) tabl head (triangle) for A164659 n\m 1 2 3 4 5 6 7 8 9 10 ... 0 1 0 0 0 0 0 0 0 0 0 1 1 2 0 0 0 0 0 0 0 0 2 1 1 3 0 0 0 0 0 0 0 3 1 2 1 1 0 0 0 0 0 0 4 1 1 3 1 5 0 0 0 0 0 5 1 2 1 1 1 3 0 0 0 0 6 1 1 1 1 5 1 7 0 0 0 7 1 2 1 1 1 3 1 1 0 0 8 1 1 3 1 1 1 7 1 9 0 9 1 2 1 1 1 1 1 1 1 5 . . . Row sums give A164663: [1, 3, 5, 5, 11, 9, 17, 11, 25, 15, 31, 21, 35, 25, 49, 23, 55, 29, 53, 39, 71, 41, 77, 43, 79,...] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Triangle of rationals A164658/A164659: a(n,m) tabl head (triangle) for A164658(n,m)/A164659(n,m) n\m 1 2 3 4 5 6 7 8 9 10 ... 0 1 1 0 1/2 2 -1 0 2/3 3 0 -3/2 0 1 4 1 0 -8/3 0 8/5 5 0 5/2 0 -5 0 8/3 6 -1 0 6 0 -48/5 0 32/7 7 0 -7/2 0 14 0 -56/3 0 8 8 1 0 -32/3 0 32 0 -256/7 0 128/9 9 0 9/2 0 -30 0 72 0 -72 0 128/5 . . . Row sums give A164660(n)/A164661(n)=:r(n). [1, 1/2, -1/3, -1/2, -1/15, 1/6, -1/35, -1/6, -1/63, 1/10, -1/99, -1/10, -1/143, 1/14, -1/195, -1/14, -1/255, 1/18, -1/323, -1/18, -1/399, 1/22, -1/483, -1/22, -1/575, 1/26, -1/675, -1/26, -1/783, 1/30, -1/899, -1/30, -1/1023, 1/34, -1/1155, -1/34, -1/1295, 1/38, -1/1443, -1/38, -1/1599,...]. Conjecture: r(n)= 1 if n=0, if n is even r(n)= -1/((n-1)*(n+1)) and if n is odd r(n)=((-1)^((n-1)/2))/(2*(2*floor((n-1)/4)+1)). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% e.o.f.%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%