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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.

%I A337760 #21 Sep 21 2020 04:58:08
%S A337760 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,
%T A337760 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,
%U A337760 0,0,2,0,1,0,1,0,0,0,-1,0,-1,0,0,0,-1,0,0
%N 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.
%C A337760 This coefficients appear in Euler totient function exact formula.
%F A337760 T(1, 1) = 1,
%F A337760 T(n, r) = 0 if r < 0 or r > n*(n+1)/2,
%F A337760 T(n, 0) = T(n - 1, n) if n is even,
%F A337760 T(n, 0) = 0 if n is odd,
%F A337760 T(n, r)  = T(n - 1, n - r) + (-1)^n*(T(n - 1, n + r) - T(n - 1, r - n)).
%e A337760 sin(t) = sin(t),
%e A337760 2*sin(t)*sin(2*t) = cos(t)-cos(3*t),
%e A337760 4*sin(t)*sin(2*t)*sin(3*t) = sin(2*t)+sin(4*t)-sin(6*t),
%e A337760 8*sin(t)*sin(2*t)*sin(3*t)*sin(4*t) = 1-cos(6*t)-cos(8*t)+cos(10*t),
%e A337760 ...
%e A337760 and corresponding table is:
%e A337760 0, 1
%e A337760 0, 1, 0, -1
%e A337760 0, 0, 1,  0, 1, 0, -1
%e A337760 1, 0, 0,  0, 0, 0, -1,  0, -1, 0, 1
%e A337760 0, 1, 0,  1, 0, 1,  0,  0,  0, 0, 0, -1, 0, -1, 0, 1
%e A337760 0, 1, 0,  1, 0, 0,  0, -2,  0, 0, 0, -1, 0,  0, 0, 0, 0, 1, 0, 1, 0, -1
%e A337760 ...
%p A337760 an := proc (n, r) option remember;
%p A337760 if n < 0 or r < 0 then
%p A337760 0
%p A337760 elif n = 1 then
%p A337760 if r = 1 then
%p A337760 1
%p A337760 else
%p A337760 0
%p A337760 end if;
%p A337760 elif r=0 and n mod 2 = 0 then
%p A337760 procname(n-1, n-r)
%p A337760 else
%p A337760 procname(n-1, n-r)+(-1)^n*(procname(n-1, n+r)-procname(n-1, r-n))
%p A337760 end if
%p A337760 end proc
%t A337760 Table[Expand[2^(n-1)*TrigReduce[Product[Sin[k*t],{k,1,n}]]],{n,1,10}]
%K A337760 sign,tabf
%O A337760 1,48
%A A337760 _Gevorg Hmayakyan_, Sep 18 2020