cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A277170 Numerator of 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 1).

Original entry on oeis.org

1, -1, 1, -1, 1, -3, 1, -1, 25, -1, 1, -49, 1, -1, 9, -3, 1, -363, 3025, -1, 169, -169, 1, -3, 1, -49, 289, -289, 7225, -361, 361, -361, 1, -1, 1, -529, 529, -529, 330625, -148225, 3025, -675, 9, -3, 7569, -2523, 142129, -409757907, 808201, -961, 8649, -2883, 1, -147
Offset: 0

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Author

Seiichi Manyama, Oct 19 2016

Keywords

Comments

Neil Calkin found the closed forms of 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 1) in 2007.

References

  • Jonathan Borwein, David Bailey, Mathematics by Experiment, 2nd Edition: Plausible Reasoning in the 21st Century.

Links

Crossrefs

Cf. A005810, A066802, A052203, A187364, A277520.

Programs

  • Mathematica
    a[n_] := HypergeometricPFQ[{3n, -n, n+1}, {2n+1, n+1/2}, 1] // Numerator; Table[a[n], {n, 0, 53}] (* Jean-François Alcover, Oct 22 2016 *)

Formula

(s(n) =) 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 1) = a(n) / A277520(n).
s(2k) = (A005810(k) / A066802(k))^2 = (((4k)! * (3k)!) / ((6k)! * k!))^2.
s(2k+1) = -1/3 * (A052203(k) / A187364(k))^2 = -1/3 * (((4k+1)! * (3k)!) / ((6k+1)! * k!))^2.
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