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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Number of odd binomial coefficients
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by Heiko Harborth PDF
Proc. Amer. Math. Soc. 62 (1977), 19-22 Request permission

Abstract:

Let $F(n)$ denote the number of odd numbers in the first n rows of Pascal’s triangle, and $\theta = (\log 3)/\log 2)$. Then $\alpha = \lim \sup F(n)/{n^\theta } = 1$, and $\beta = \lim \inf F(n)/{n^\theta } = 0.812\;556\; \ldots .$
References
  • N. J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly 54 (1947), 589–592. MR 23257, DOI 10.2307/2304500
  • Heiko Harborth, Über die Teilbarkeit im Pascal-Dreieck, Math.-Phys. Semesterber. 22 (1975), 13–21 (German). MR 384676
  • David Singmaster, Notes on binomial coefficients. III. Any integer divides almost all binomial coefficients, J. London Math. Soc. (2) 8 (1974), 555–560. MR 396285, DOI 10.1112/jlms/s2-8.3.555
  • K. B. Stolarsky, Digital sums and binomial coefficients, Notices Amer. Math. Soc. 22 (1975), A-669. Abstract #728-A7.
  • Kenneth B. Stolarsky, Power and exponential sums of digital sums related to binomial coefficient parity, SIAM J. Appl. Math. 32 (1977), no. 4, 717–730. MR 439735, DOI 10.1137/0132060
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 62 (1977), 19-22
  • MSC: Primary 10A20
  • DOI: https://doi.org/10.1090/S0002-9939-1977-0429714-1
  • MathSciNet review: 0429714