The consistently intriguing posts of The Everything Seminar has landed it on the sidebar. The latest post about summing divergent series includes the generating function

$T(z) = 1 + z + 2 z^2 + 5 z^3 + 14 z^4 + 42 z^5 + \cdots$

for the Catalan numbers. M theorists will recognise the coefficients as the number of vertices on an associahedron: two vertices for a line segment, five for a pentagon, and so on. And lest there be any doubt he is thinking about trees, he refers to this paper, which maps 7-tuples of (planar rooted binary) trees to trees. A commenter (with a blog called God Plays Dice) pointed out that a different set of trees gave another isomorphism between 5-tuples and singlets.

Apparently the reason the number 7 works for general trees is because 7 = 6 mod 1 and by dividing trees into left and right halves we get an equation $T = 1 + T^2$, where the 1 stands for the root. So planar rooted trees are associated to a sixth root of unity, and the fourth root case is about trees with either 0, 1 or 2 children at each vertex. Its generating function $M$ yields the series

$M(1) = 1 + 2 + 4 + 9 + 21 + 51 + · · · = −i$

It seems there are lots of cute ways of writing down complex numbers as infinite sums, so long as one uses series derived from trees! Here’s a cool TWF on this stuff, with a link to this helpful, and seminal, paper by Fiore and Leinster. Oh, I can’t wait to go and play some more…