A337760 Irregular triangle where T(n,k) are the coefficients of expansion 2^(n-1) Product_{k=1..n} sin(k*t) = Sum_{k=1..n*(n+1)/2} T(n,k)*cos(k*t) for even n and 2^(n-1) Product_{k=1..n} sin(k*t) = Sum_{k=1..n*(n+1)/2} T(n,k)*sin(k*t) for odd n.
0, 1, 0, 1, 0, -1, 0, 0, 1, 0, 1, 0, -1, 1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 1, 0, 1, 0, 1, 0, 0, 0, -2, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 1, 0, -1, 0, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 0
Offset: 1
Examples
sin(t) = sin(t), 2*sin(t)*sin(2*t) = cos(t)-cos(3*t), 4*sin(t)*sin(2*t)*sin(3*t) = sin(2*t)+sin(4*t)-sin(6*t), 8*sin(t)*sin(2*t)*sin(3*t)*sin(4*t) = 1-cos(6*t)-cos(8*t)+cos(10*t), ... and corresponding table is: 0, 1 0, 1, 0, -1 0, 0, 1, 0, 1, 0, -1 1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 1 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 1 0, 1, 0, 1, 0, 0, 0, -2, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 1, 0, -1 ...
Programs
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Maple
an := proc (n, r) option remember; if n < 0 or r < 0 then 0 elif n = 1 then if r = 1 then 1 else 0 end if; elif r=0 and n mod 2 = 0 then procname(n-1, n-r) else procname(n-1, n-r)+(-1)^n*(procname(n-1, n+r)-procname(n-1, r-n)) end if end proc
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Mathematica
Table[Expand[2^(n-1)*TrigReduce[Product[Sin[k*t],{k,1,n}]]],{n,1,10}]
Formula
T(1, 1) = 1,
T(n, r) = 0 if r < 0 or r > n*(n+1)/2,
T(n, 0) = T(n - 1, n) if n is even,
T(n, 0) = 0 if n is odd,
T(n, r) = T(n - 1, n - r) + (-1)^n*(T(n - 1, n + r) - T(n - 1, r - n)).
Comments