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Irregular triangle read by rows (see examples). Consider an arbitrary anharmonic oscillator with Hamiltonian energy: H=(1/2)*b^2=(1/2)*(p^2+q^2) + Sum_{i=3} 2*v_i*q^i, and a stable minimum at (p,q)=(0,0). The phase space trajectory can be written in polar phase space coordinates as (q,p) = (R(x)cos(x),R(x)sin(x))=(R(Q)Q,R(Q)P). The present triangle determines a power / Fourier series of R(Q): R(Q) = b * (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 constructed from partitions of "n" listed in reverse lexicographic order. Partition n=(z_1+z_2+...z_j) becomes 2*Q^((z_1+2)+(z_2+2)+...(z_j+2))*v_{z_1+2}*v_{z_2+2}*...*v_{z_j+2} (see examples). This sequence transforms into A273506/A273507 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 phase space trajectory A276738 has phase space angular velocity A276814 and differential time dependence A276815. We calculate the period K = Int dt over the range [2*Pi, 0], trivial to compute from A276815 using A273496. Then K/(2*Pi) = 1 + sum b^(2n)*T(n,m)*f'(n,m); where the sum runs over n = 1, 2, 3 ... and m = 1, 2, 3, ... A000041(2n), and f'(n,m) = f(2n,m) of A276738 with Q=1/2. Choosing one point from the infinite dimensional coefficient space--v_i=0 for odd i, v_i=(-1)^(i/2-1)/2/(i!) otherwise--setting b^2 = 4*k, and summing over the entire table obtains the EllipticK expansion 2*A038534/A038533. For more details read "Plane Pendulum and Beyond by Phase Space Geometry" (Klee, 2016).

Irregular triangle read by rows (see examples).

Consider an axially symmetric oscillator in two dimensions with polar coordinates ( r, y ). By conservation of angular momentum, replace the cyclic angle coordinate y with dy/dt = 1/r^2. The system becomes one-dimensional in r, with an effective potential including the 1/r^2 term. Assume that the effective potential has a minimum around r0 and apply a linear transform r --> q = r-r0. Radial oscillations around the effective potential minimum follow the exact solution of A276738, A276814, A276815, A276816. Now dy = dx (dy/dt) / (dx/dt) = dx * Sum b^n*T(n,m)*F(n,m), with n=1,2,3.... and m=1,2,3...A000070(n). Basis functions F(n,m) are an ordered union over A276738's f(n,m): F(n,m')={ (1/r0^2)*(Q/r0)^n } & Append_{i=1..n}_{m=1..A000041(n)} (1/2/r0^2)*(Q/r0)^(n - i)*f(i,m), where each successive term f(i,m) is appended such that index m' inherets the ordering of each m index (see examples). Integrating dx over a range of 2 Pi loses all odd rows, as in A276815 / A276816. This sequence is a useful tool in classical and relativistic astronomy (follow links to Wolfram demonstrations).

Contribution from Wolfdieter Lang, Nov 08 2010: (Start)

a(n)/A056982(n) = -(binomial(2*n,n)^2)/((2*n-1)*2^(4*n)), n>=0, are the coefficients of x^n of hypergeometric([1/2,-1/2],[1],x).

The series hypergeometric([1/2,-1/2],[1],e^2)=L/(2*Pi*a) with L the perimeter of an ellipse with major axis a and numerical eccentricity e. (End)

Equivalently, denominators in partial products of the following approximation to Pi: Pi = Product_{n>=1} 4*n^2/(4*n^2-1). Numerators are in A056982.

If one takes the row polynomials as R(n, x) = Sum_{m=0..n} T(n, m)*x^(2*m+1), n >= 0, Jacobi's elliptic sn(u|k) function in terms of the new variables v and q becomes sn(u|k) = Sum_{n>=0} R(n, x)*q^n, if one replaces in R(n, x) x^j by sin(j*v).

v=v(u,k^2) and q=q(k^2) are computed with the help of A038534/A056982 for (2/Pi)*K and A002103 for q expanded in powers of (k/4)^2.

A test for sn(u|k) with u = 1, k = sqrt(1/2), that is v approximately 0.8472130848 and q approximately 0.04321389673, with rows n=0..10 (q powers not exceeding 10) gives 0.8030018002 to be compared with sn(1|sqrt(1/2)) approximately 0.8030018249.

For the derivation of the Fourier series formula of sn given in Abramowitz-Stegun (but there the notation sn(u|m=k^2) is used for sn(u|k)) see, e.g., Whittaker and Watson, p. 511 or Armitage and Eberlein, Exercises on p. 55.

For the cn expansion see A274661.

See also the W. Lang link, equations (34) and (35).

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).

This is the instance m=1/2 of the partial sums r(m,n) = Sum_{k=0..n} (risefac(m,k)/ k!)^2, where risefac(x,k) = Product_{j=0..k-1} (x+j), and risefac(x,0) = 1.

The limit n -> oo does not exist. It would be hypergeometric([1/2,1/2],[1],z -> 1), which diverges.

The partial sums of the cubes converge for |m| < 2/3. See Morley's series under A277232 (for m=1/2).

a(n)/A056982(n) are the Landau constants. These constants are defined as G(n) = Sum_{j=0..n} g(j)^2, where g(n) = (2*n)!/(2^n*n!)^2 = A001790(n)/A046161(n). - Peter Luschny, Sep 27 2019

For the denominators see A274654.

The coefficients of z^n for the expansion of F_1(1/2,1/2;z) are A274655(n)/A274656(n).

Fricke's hypergeometric function F_1(a,b;z) = Sum_{n > = 0} f(a,b;n)*z^n/n!, satisfies the recurrence

f(a,b,n) = ((a+n-1)*(b+n-1)/n)*f(a,b;n-1) + c(a,b;n)*(1/(a+n-1) + 1/ (b+n-1) - 2/n), with c(a,b;n) = [z^n/n!]hypergeometric([a,b],[1],z) = risefac(a,n) * risefac(b,n)/n!, where risefac is the rising factorial (Pochhammer's symbol) and the input is f(a,b;0)= 0. See the Fricke I reference, p. 114.

The hypergeometric function F_1(1/2,1/2;z) appears in the formula for (2/Pi) K'(k) + (1/Pi)*log(k^2/16)*(2/Pi)*K(k) = F_1(1/2,1/2;k^2), where K and sqrt(-1)*K' are the real and imaginary quarter periods, and k is the modulus (k^2 is the parameter) of elliptic functions. See the Fricke I reference p. 465, eq. (11), and also Fricke III, p. 2, eq. (3).

(2/Pi)*K(k) = hypergeometric([1/2,1/2],[1],k^2). For the expansion coefficients see A038534/A056982 and also A274657/A123854.