*N.B. A detailed on-line essay by S. Finch
was the starting point for this entry.*

Let be the number of 1s in the Binary expression of . Then the number of Odd Binomial
Coefficients where is (Glaisher 1899, Fine 1947). The
number of Odd elements in the first rows of Pascal's Triangle is

(1) |

(2) |

(3) |

(4) |

**References**

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/stlrsky/stlrsky.html

Fine, N. J. ``Binomial Coefficients Modulo a Prime.'' *Amer. Math. Monthly* **54**, 589-592, 1947.

Wolfram, S. ``Geometry of Binomial Coefficients.'' *Amer. Math. Monthly* **91**, 566-571, 1984.

© 1996-9

1999-05-26