A248097 Number of ordered trees with n edges such that non-leaf vertices at even (odd) level have an even (odd) number of children.

1, 0, 1, 2, 2, 8, 15, 34, 93, 208, 548, 1390, 3527, 9318, 24203, 64052, 170392, 454020, 1220651, 3288010, 8897474, 24170066, 65829806, 179901956, 492872483, 1353655974, 3726557183, 10280008874, 28415211437, 78686254872, 218265740153, 606417214412, 1687349373815

Offset

0,4

Links

Table of n, a(n) for n=0..32.

Formula

G.f. h(z) is given by h = 1/(1-z^2*g^2), where g = 1 + z*h/(1-z^2*h^2). The function g(z) is the g.f. of the companion sequence A248096.

G.f. h(z) is given by h*((1 - z^2*h^2)^2 - z^2*(1 + z*h - z^2*h^2)^2) = (1 - z^2*h^2)^2.

Example

a(4) = 2 because we have (i) the path-trees P[2] and P[3] joined at their endpoints and (ii) the path-trees P[3] and P[2] joined at their endpoints.

Maple

eq:= h = 1/(1-z^2*g^2): g:= 1+z*h/(1-z^2*h^2): h := RootOf(eq, h): hser := series(h, z=0, 35): seq(coeff(hser, z, n), n=0..32);

Mathematica

max=40; s[0]={}; s[n_] := s[n] = Join[s[n-1], h=Sum[a[k]*z^k, {k, 0, n}] /. s[n-1]; SolveAlways[h == Normal[Series[1/(1-z^2*g^2) /. g -> 1+z*h/(1-z^2*h^2), {z, 0, n}]], z] // First]; Table[a[n] /. s[n+1], {n, 0, max}] (* Jean-François Alcover, Dec 26 2014, translated and adapted from Maple *)

Crossrefs

Cf. A248096.

Sequence in context: A363181 A349648 A295193 * A098273 A342835 A361294

Adjacent sequences: A248094 A248095 A248096 * A248098 A248099 A248100

Keyword

nonn

Author

Emeric Deutsch, Dec 25 2014

Status

approved