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 A224873 Triangle of coefficients T(n, k) (n > 0, k < n) in expansion of sin(u*x) * sin(v*x) / (cos(u*x) + cos(v*x)), read by rows. 0

%I #11 Jul 24 2013 12:52:24

%S 1,1,1,3,25,3,17,329,329,17,155,5325,14301,5325,155,2073,110605,

%T 563013,563013,110605,2073,38227,2918825,23904881,45956625,23904881,

%U 2918825,38227,929569,96075665,1150348017,3600524785,3600524785,1150348017,96075665,929569

%N Triangle of coefficients T(n, k) (n > 0, k < n) in expansion of sin(u*x) * sin(v*x) / (cos(u*x) + cos(v*x)), read by rows.

%H Michael Hardy, <a href="http://mathoverflow.net/questions/116229/">A "known" tangent half-angle formula?</a>.

%F A024235(n) = Sum_{k = 0..n-1} T(n, k).

%F A110501(n) = T(n, 0).

%F E.g.f.: sin(u*x) * sin(v*x) / (cos(u*x) + cos(v*x)) = Sum_{n>0, k<n} x^(2*n) / (2*n)! * u^(2*k + 1) * v^(2*n - 2*k -1) * T(n, k).

%e 1; 1, 1; 3, 25, 3; 17, 329, 329, 17; ...

%o (PARI) {T(n, k) = local(u = 'u, v = 'v, A); if( n<0 || k>=n, 0, n = 2*n; k = 2*k + 1; A = x * O(x^n); n! * polcoeff( polcoeff( polcoeff( sin( u*x + A) * sin( v*x + A) / (cos( u*x + A) + cos( v*x + A)), n, x), k, u), n - k, v))}

%Y Cf. A024235, A110501.

%K nonn,tabl

%O 1,4

%A _Michael Somos_, Jul 23 2013

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Last modified September 13 09:21 EDT 2024. Contains 375904 sequences. (Running on oeis4.)