An oddity requiring an explanation...

[Gordon Royle]

Here is something odd to ponder.

Let C_n(v) denote the polynomial that counts the spanning connected  
subgraphs of K_n according to the number of edges...

For example, C_4(v) = 16 v^3 + 15 v^4 + 6 v^5 + v^6  because K_4 has 16  
spanning trees, 16 spanning subgraphs with 4 edges, 6 spanning  
subgraphs with 5 edges and 1 spanning subgraph with 6 edges.

Now consider the sequence

	C_3(-2), C_4(-2), C_5(-2), C_6(-2), C_7(-2) ...

which is

	4, -16, 80, -512, 3904, ....


and look it up on the OEIS....

http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/ 
eisA.cgi?Anum=A000831


If we ignore the signs we discover that this sequence is actually the  
series of coefficients of the expansion of (1+tan(x))/(1-tan(x)) =  
tan(x+Pi/4) when we view this as an exponential generating function...


In other words

(1-tan(x))/(1+tan(x)) = tan(x + Pi/4) = 1 + 2 x + 4 / (2!) x^2 + 16 /  
(3!) x^3 + 80 / (4!) x^4 + 512 / (5!) x^5 + 3914 / (6!) x^6 + 34816 /   
(7!) x^7 .....



This is not just a low-order coincidence as it continues for as many  
coefficients as I can compute....


Any explanations?

Gordon