L. Carlitz, A problem in partitions related to the Stirling numbers, Bull. Amer. Math. Soc. 70, 275-278 (1964).

If one denotes by S(n,r) the Stirling numbers of second kind, then the number theta(n) of odd S(n+1,2r+1), 0<=2n<r satisfies the generating function prod(k>=0, 1+x^(2^k)+x^(2^(k+1))).

It follows that theta(n) is the number of partitions n=n_0+2n_1+4n_2+8n_3+... (0<=n_j<=2).

The author gives numerous recurrence formulas for theta(n), from which follow continued fraction and determinant representations.

Solutions of theta(x)=t are examined, and the behaviour of theta(n) for large n is discussed.

[R. Stephan, after H. Salié]