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A246543
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a(n) = (2/n^3)*( Sum_{k=0..n-1} (-1)^k*(3*k^2+3*k+1)*binomial(n-1,k)^3*binomial(n+k,k)^3 ).
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6
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2, -47, 1142, 3793, -4094806, 371557891, -13021558306, -1374157073639, 281067953420114, -22220280272696387, -51611579093593498, 257837341935815261683, -35155217354672369625958, 1761633462267526777842223, 202464167122130621896038062
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OFFSET
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1,1
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COMMENTS
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Conjecture: Let n be any positive integer. For m = 0, 2, 4, ..., we have Sum_{k=0..n-1} (3k^2+3k+1)*(binomial(n-1,k)*binomial(n+k,k))^m == 0 (mod n^3); for m = 1, 3, 5, ... we have 2*Sum_{k=0..n-1} (-1)^k*(3k^2+3k+1)*(binomial(n-1,k)*binomial(n+k,k))^m == 0 (mod n^3).
The Zeilberger algorithm could yield a complicated fifth-order recurrence for a(n).
The author proved the conjecture in the latest version of arXiv:1408.5381. - Zhi-Wei Sun, Sep 14 2014
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LINKS
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EXAMPLE
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a(2) = -47 since (2/2^3)*( Sum_{k=0..1} (-1)^k*(3k^2+3k+1)*binomial(1,k)^3*binomial(2+k,k)^3 ) = (1/4)*(1-7*3^3) = -47.
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MATHEMATICA
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a[n_] := Sum[(3 k^2 + 3 k + 1) (-1)^k (Binomial[n - 1, k] Binomial[n + k, k])^3, {k, 0, n - 1}] 2/n^3
Table[a[n], {n, 1, 14}]
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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