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Irregular triangle read by rows ( see examples ). The phase space trajectory of A276738 has one time dependent variable, the phase space angle "x" defined as Tan[x]=p/q. Then dx/dt = cos[x]^2* d/dt(p/q), which can be written as a function of Q=cos[x] by application of the classical equations of motion d/dt(p,q) = ( -d/dq H, d/dp H ), with H the anharmonic oscillator Hamiltonian. Substituting the result of A276738 and expanding in powers of b, we obtain dx/dt = -1 + sum b^n*T(n,m)*f(n,m); where the sum runs over n=1,2,3... and m = 1,2,3, ... A000041(n). The basis functions f(n,m) are the same as in A276738. Observe the limit where Q --> 0, dx/dt --> -1, the harmonic oscillator value. Similarly if v_i --> 0 then dx/dt --> -1.

The phase space trajectory A276738 has phase space angular velocity A276814, which allows expansion of dt = dx /(dx/dt) = dx(-1 + sum b^n*T(n,m)*f(n,m)); where the sum runs over n = 1, 2, 3 ... and m = 1, 2, 3, ... A000041(n). The basis functions f(n,m) are the same as in A276738. To obtain period K, we integrate the function of Q=cos[x] over a range of [2*pi,0]. All odd powers of Q integrate to zero, so the period is an expansion in E=(1/2)*b^2 (Cf. A276816). This sequence transforms into A274076/A274078 by setting v_i=0 for odd i, v_i=(-1)^(i/2-1)/2/(i!) otherwise, and (1/2)*b^2 = 2*k. For more details read "Plane Pendulum and Beyond by Phase Space Geometry" (Klee, 2016).

The denominator triangle is given in A273168.

The expansion is x^(2*n) = Sum_{m=0..n} R(n, m)*Tnx(2*m, x), n >= 0, with the rational triangle R(n, m) = a(n, m)/A273168(n, m).

Compare this with A127673.

This is equivalent to the expansion cos(x)^(2n) = Sum_{m=0..n} R(n, m)*cos(2*m*x), n >= 0. Compare this with the even numbered rows of A273496.

See A244420/A244421 for the expansion of x^(2*n+1) in terms of odd-indexed Chebyshev polynomials of the first kind.

The signed rational triangle S(n, m) = R(n, m) * (-1)^m appears in the expansion sin(x)^(2n) = Sum_{m=0..n} S(n, m) * cos(2*m*x), n >= 0. This is equivalent to the identity (1-x^2)^n = Sum_{m=0..n} S(n, m) * T(2*m, x).

The triangle for the denominators is given in A273172.

Int(cos^(2*n+1)(x), x) = Sum_{m = 0..n} R(n, m)*sin((2*m+1)*x), n >= 0, with the rational triangle a(n, m)/A273172(n, m).

For the rational triangle for the odd powers of cos see A244420/A244421. See also the odd-indexed rows of A273496.

The signed rational triangle S(n, m) = R(n, m)*(-1)^(m+1) appears in the formula

Int(sin^(2*n+1)(x), x) = Sum_{m = 0..n} S(n, m)*cos((2*m+1)*x), n >= 0,

The denominator triangle is given in A273170.

Int(cos^(2*n)(x), x) = R(n, 0)*x + Sum_{m = 1..n} R(n, m)*sin(2*m*x), n >= 0, with the rational triangle a(n, m)/A273170(n, m).

For the rational triangle for the even powers of cos see A273167/A273168. See also the even-indexed rows of A273496.

For the integral over odd powers of cos see the rational triangle A273171/A273172.

The signed triangle S(n, m) = R(n, m)*(-1)^m appears in the integral of even powers of sin as Int(sin^(2*n)(x), x) = S(n , 0)*x + Sum_{m = 1..n} S(n, m)*sin(2*m*x), n >= 0.