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A211601
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a(n) = (binomial(p^n, p^(n-1)) - binomial(p^(n-1), p^(n-2))) / p^(3n-2) for p = 3.
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1
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OFFSET
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2,2
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COMMENTS
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Consider the difference between two binomials f(p,k) = binomial(p^k, p^(k-1)) - binomial(p^(k-1), p^(k-2)).
A theorem from the A. I. Shirshov paper (in Russian) states:
p^(3k - 3) divides f(p,k) for prime p = 2 and k > 2.
p^(3k - 2) divides f(p,k) for prime p = 3 and k > 1.
p^(3k - 1) divides f(p,k) for prime p > 3 and k > 1.
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REFERENCES
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D. B. Fuks and Serge Tabachnikov, Mathematical Omnibus: Thirty Lectures on Classic Mathematics, American Mathematical Society, 2007. Lecture 2. Arithmetical Properties of Binomial Coefficients, pages 27-44
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LINKS
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Table of n, a(n) for n=2..5.
D. B. Fuks and M. B. Fuks, Arithmetics of binomial coefficients, Kvant 6 (1970), 17-25. (in Russian)
A. I. Shirshov, On one property of binomial coefficients, Kvant 10 (1971), 16-20. (in Russian)
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FORMULA
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a(n) = ((binomial(3^n, 3^(n-1)) - binomial(3^(n-1), 3^(n-2))) / 3^(3n-2).
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MATHEMATICA
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p = 3; Table[(Binomial[p^n, p^(n - 1)] - Binomial[p^(n - 1), p^(n - 2)]) / 3^(3n - 2), {n, 2, 6}]
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CROSSREFS
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Cf. A211600, A211602.
Sequence in context: A116095 A347040 A252547 * A199745 A200145 A179271
Adjacent sequences: A211598 A211599 A211600 * A211602 A211603 A211604
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KEYWORD
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nonn
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AUTHOR
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Alexander Adamchuk, Apr 16 2012
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STATUS
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approved
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