The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A246512 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A246512

a(n) = (sum_{k=0}^{n-1}(3k^2+3k+1)*C(n-1,k)^2*C(n+k,k)^2)/n^3, where C(n,k) denotes the binomial coefficient n!/(k!*(n-k)!).
(history; published version)

#29 by Michael De Vlieger at Fri Dec 24 08:12:14 EST 2021

STATUS

reviewed

approved

#28 by Joerg Arndt at Fri Dec 24 02:28:40 EST 2021

STATUS

proposed

reviewed

#27 by Michel Marcus at Fri Dec 24 00:18:05 EST 2021

STATUS

editing

proposed

#26 by Michel Marcus at Fri Dec 24 00:17:57 EST 2021

LINKS

Zhi-Wei Sun, <a href="http://arxiv.org/abs/1408.5381">Two new kinds of numbers and related divisibility results</a>, arXiv:1408.5381, [math.NT], 2014-2018.

Zuo-Ru Zhang, <a href="https://arxiv.org/abs/2112.12427">Proof of two conjectures of Z.-W. Sun on combinatorial sequences</a>, arXiv:2112.12427 [math.CO], 2021.

PROG

(PARI) a(n) = sum(k=0, n-1, (3*k^2+3*k+1)*binomial(n-1, k)^2*binomial(n+k, k)^2) /n^3; \\ Michel Marcus, Dec 24 2021

STATUS

approved

editing

#25 by Bruno Berselli at Wed Feb 11 06:34:47 EST 2015

STATUS

proposed

approved

#24 by Jon E. Schoenfield at Wed Feb 11 06:02:19 EST 2015

STATUS

editing

proposed

#23 by Jon E. Schoenfield at Wed Feb 11 06:02:14 EST 2015

COMMENTS

In the latest version of arXiv:1408:5381, the author proved that a(n) is always an integer. Notice that a(65) is relatively prime to 65. - Zhi-Wei Sun, Sep 14, 2014

EXAMPLE

a(2) = 8 since sum_{k=0,1} (3k^2+3k+1)C(1,k)^2*C(2+k,k)^2 = 1 + 7*3^2 = 64 = 2^3*8.

STATUS

approved

editing

#22 by Michael Somos at Sun Sep 14 19:56:04 EDT 2014

STATUS

proposed

approved

#21 by Zhi-Wei Sun at Sun Sep 14 19:11:46 EDT 2014

STATUS

editing

proposed

#20 by Zhi-Wei Sun at Sun Sep 14 19:11:40 EDT 2014

COMMENTS

In the latest version of arXiv:1408:5381, the author proved that a(n) is always an integer. Notice that a(65) is relatively prime to 65. - _ Zhi-Wei Sun_, Sep 14, 2014

STATUS

approved

editing