cp's OEIS Frontend

Terms are always odd. a(1) = A061548(2), a(2) = A061548(3), a(3) = A061548(5), a(4) = A061548(9), a(5) = A061548(17), ... Hence it seems that a(n) = A061548(A000051(n)).

C(2*k, k)/2 = C(2*k-1, k) = C(2*k-1, k-1) is odd if and only if k = 2^n. - Michael Somos, Mar 12 2014

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.

Also the numerators of the nonzero coefficients in the expansion of hypergeom([Seq_{k=1..m-1} k/m], [], (x/m)^m) for m = 3.

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

All even coefficients of t_n have to be 0, because t_n is defined to be point-symmetric with respect to the origin, with vanishing n-th derivative for x=1.

A sigmoidal transfer function sigma_n: R->[ -1,1] can be defined as sigma_n(x) = 1 if x>1, sigma_n(x) = t_n(x) if x in [ -1,1] and sigma_n(x) = -1 if x<-1.

Positions where A056616 differs from A000984.

The set seems to have greater cardinality than its complement.

Positions where A055786 differs from A001790. - Mohammed Yaseen, Aug 03 2024

Numerators of one half of norm square of monic Legendre polynomials p_n(x).

The denominators of these polynomials are given by A069955.

The rationals N2(n) = 2*a(n)/A069955(n) give the minimal norm square for real monic polynomials. The norm square is defined as integral over the interval [-1,+1] of the square of the polynomials. Cf. the Courant-Hilbert reference.