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A211600
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a(n) = (binomial(p^n, p^(n-1)) - binomial(p^(n-1), p^(n-2))) / p^(3n-3) for p = 2.
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2
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OFFSET
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3,2
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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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FORMULA
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a(n) = (binomial(2^n, 2^(n-1) - binomial(2^(n-1), 2^(n-2))) / 2^(3n-3).
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EXAMPLE
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a(3) = 1 is the difference between central binomials C(8,4) - C(4,2) = 70 - 6 = 64 divided by 2^(3*2 - 3) = 64.
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MAPLE
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MATHEMATICA
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p = 2; Table[(Binomial[p^n, p^(n - 1)] - Binomial[p^(n - 1), p^(n - 2)]) / 2^(3n - 3), {n, 3, 9}]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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