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Number of odd binomial coefficients. (English) Zbl 0323.10043

Summary: 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 \( \beta = \lim \inf F(n)/{n^\theta } = 0.812\;556\; \ldots \).

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
05A10 Factorials, binomial coefficients, combinatorial functions
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References:

[1] N. J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly 54 (1947), 589 – 592. · Zbl 0030.11102
[2] Heiko Harborth, Über die Teilbarkeit im Pascal-Dreieck, Math.-Phys. Semesterber. 22 (1975), 13 – 21 (German). · Zbl 0296.10009
[3] David Singmaster, Notes on binomial coefficients. III. Any integer divides almost all binomial coefficients, J. London Math. Soc. (2) 8 (1974), 555 – 560. · Zbl 0293.05007
[4] K. B. Stolarsky, Digital sums and binomial coefficients, Notices Amer. Math. Soc. 22 (1975), A-669. Abstract #728-A7.
[5] 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. · Zbl 0355.10012
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