cp's OEIS Frontend

Even denominators of coefficients in Taylor series expansion of 2 - 2*cos(x) - 2*x*sin(x) + x^2.

Equivalent to the even denominators of expansion of (1-cos(x))^2 + (x-sin(x))^2, which is the square of the secant length measured from the origin (0,0) to the cycloid point (1-cos(x), x-sin(x)). Note that only x^4 has the first nonzero coefficient of the series.

Numerators of the Taylor series expansion are given by A327883.

The Taylor series itself has an expansion Sum_{k>=2} (-1)^k*2*(2*k-1)/(2*k)!*x^(2*k).

Here, a star graph is a tree on n nodes (n>=4) with one specially designated (center) vertex, v of degree n-1. We are allowed to add edges so that the degree of any node (excepting v) is at most 3 and so that every cycle includes the vertex v with the possible exception of a single cycle of length n-1 through each non-center vertex. We note that anytime edges are added the original "center" node remains specially designated. a(n) is the number of such connected simple labeled graphs with a specially designated node.

If we don't add any edges we have a star graph and there are n labelings.

If we add exactly one edge then we produce a cycle of length 3 and we no longer have a tree.

If we add the maximum number of edges we get a wheel graph A171005.