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.

1, 1, 1, 3, 25, 3, 17, 329, 329, 17, 155, 5325, 14301, 5325, 155, 2073, 110605, 563013, 563013, 110605, 2073, 38227, 2918825, 23904881, 45956625, 23904881, 2918825, 38227, 929569, 96075665, 1150348017, 3600524785, 3600524785, 1150348017, 96075665, 929569

Offset

1,4

Links

Table of n, a(n) for n=1..36.

Michael Hardy, A "known" tangent half-angle formula?.

Formula

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

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

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).

Example

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

Prog

(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))}

Crossrefs

Cf. A024235, A110501.

Sequence in context: A080202 A300945 A234319 * A085836 A073916 A076962

Adjacent sequences: A224870 A224871 A224872 * A224874 A224875 A224876

Keyword

nonn,tabl

Author

Michael Somos, Jul 23 2013

Status

approved