The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A000172 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A000172

The Franel number a(n) = Sum_{k = 0..n} binomial(n,k)^3.
(history; published version)

#361 by Michael De Vlieger at Sat Nov 02 09:11:35 EDT 2024

STATUS

reviewed

approved

#360 by Joerg Arndt at Sat Nov 02 07:16:14 EDT 2024

STATUS

proposed

reviewed

#359 by Peter Bala at Sat Nov 02 07:04:49 EDT 2024

STATUS

editing

proposed

#358 by Peter Bala at Thu Oct 31 18:14:42 EDT 2024

FORMULA

For n >= 1, a(n) = 2 * Sum_{k = 0..n-1} binomial(n, k)^2 * binomial(n-1, k). Cf. A361716.

#357 by Peter Bala at Thu Oct 31 18:12:16 EDT 2024

FORMULA

For n >= 1, a(n) = 2 * Sum_{k = 0..n-1} binomial(n, k)^2 * binomial(n-1, k). - _From _Peter Bala_, Oct 31 2024: (Start)

For n >= 1, a(n) = 2 * Sum_{k = 0..n-1} binomial(n, k)^2 * binomial(n-1, k).

For n >= 1, a(n) = 2 * hypergeom([-n, -n, -n + 1], [1, 1], -1). (End)

#356 by Peter Bala at Thu Oct 31 17:55:47 EDT 2024

FORMULA

For n >= 1, a(n) = 2 * Sum_{k = 0..n-1} binomial(n, k)^2 * binomial(n-1, k). - Peter Bala, Oct 31 2024

STATUS

approved

editing

#355 by Alois P. Heinz at Tue Aug 27 14:52:25 EDT 2024

STATUS

editing

approved

#354 by Alois P. Heinz at Tue Aug 27 14:23:14 EDT 2024

CROSSREFS

Column k=3 of A372307.

STATUS

approved

editing

#353 by Peter Bala at Thu Jul 18 18:59:32 EDT 2024

FORMULA

a(n) = Sum_{k = floor(n/2)..n} binomial(n, k)^2*binomial(2*k, n). - Peter Bala, Jul 18 2024

KEYWORD

nonn,easy,nice,changed

STATUS

editing

approved

#352 by Peter Bala at Thu Jul 18 18:00:02 EDT 2024

FORMULA

a(n) = Sum_{k = floor(n/2)..n} binomial(n, k)^2*binomial(2*k, n). - Peter Bala, Jul 18 2024

STATUS

approved

editing