--reminiscent of "Flanelle's theorem" Sum[ n^j * h(n), for n=0..2^k-1 ] = 0 if 0<=j<k with j=integer
(and perhaps Adam P. Goucher will inform us... who the helle was Flanelle? I think it was actually Adam P. Goucher in disguise?)
That was the name suggested by `Maria' on the cp4space article: http://cp4space.wordpress.com/2013/08/14/polynomials-and-hamming-weights/#co...
I find now a new closed form sum Sum[ C^n * h(n), for n=0..2^k-1 ] = (1-C)*(1-C^2)*(1-C^4)*...*(1-C^(2^(k-1)))
Exciting.
To prove it, you work inductively layer by layer in the "Wilson binary tree" starting from leaves. At the leaf layer, we have C^n. At the next layer, fathers of leaves, we have by differencing (1-C) * C^(2*n). At the next layer, grandfathers of leaves, we have (1-C)*(1-C^2) * C^(4*n). And so on... at the the layer k+1 above the leaves we have (1-C) * (1-C^2) * (1-C^4) * ... * (1-C^(2^k)) * C^(2^(k+1) * n) ...and now evaluate this when n=0 at the layer k-1 above leaves, QED.
You can also prove Flanelle by the same kind of layer-by-layer inductive approach, noting that differencing a polynomial of degree d, yields 0 after you do it more than d times.
That's essentially the inductive approach I used for Flanelle(P).
Note that our new sum also can be viewed as a "generating function(C)" for h(n).
Indeed, that's mentioned here: http://en.wikipedia.org/wiki/Prouhet%E2%80%93Thue%E2%80%93Morse_constant Sincerely, Adam P. Goucher