q-Powerseries and binary strings

Consider the infinite product (1+qz)(1+qz^2)(1+qz^4)(1+qz^8)\cdots, and write this as \sum\limits_{\nu \geqslant 0} q^{B_\nu}z^\nu. Relate B_\nu to the binary expression of \nu. The sequence starts off as: 1,1,2,1,2,2,3,1,2,2,3,2,3,3,4,\ldots.

q-Powerseries and binary strings

One thought on “q-Powerseries and binary strings

  1. fmartin says:

    The sequence B_n is exactly the sum of 1s appearing in the binary expression of n (one needs q not to be a root of unity in order for B_n to be well-defined).


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