The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A005810 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A005810

a(n) = binomial(4n,n).
(history; published version)

#151 by Michael De Vlieger at Tue Aug 01 09:55:38 EDT 2023

STATUS

reviewed

approved

#150 by Michel Marcus at Tue Aug 01 09:53:38 EDT 2023

STATUS

proposed

reviewed

#149 by Chai Wah Wu at Tue Aug 01 09:45:02 EDT 2023

STATUS

editing

proposed

#148 by Chai Wah Wu at Tue Aug 01 09:44:56 EDT 2023

PROG

(Python)

from math import comb

def A005810(n): return comb(n<<2, n) # Chai Wah Wu, Aug 01 2023

STATUS

approved

editing

#147 by Charles R Greathouse IV at Thu Sep 08 08:44:34 EDT 2022

PROG

(MAGMAMagma) [ Binomial(4*n, n): n in [0..100] ]; // Vincenzo Librandi, Apr 13 2011

Discussion

Thu Sep 08

08:44

OEIS Server: https://oeis.org/edit/global/2944

#146 by Alois P. Heinz at Mon Feb 21 07:20:52 EST 2022

STATUS

proposed

approved

#145 by Michel Marcus at Mon Feb 21 06:11:26 EST 2022

STATUS

editing

proposed

#144 by Michel Marcus at Mon Feb 21 06:11:23 EST 2022

LINKS

T. Tom Copeland, <a href="https://tcjpn.files.wordpress.com/2013/04/discrdeltas9-6-20122.pdf">Discriminating Deltas, Depressed Equations, and Generalized Catalan Numbers</a>, 2012, pp. 5-6.

STATUS

proposed

editing

#143 by Peter Bala at Mon Feb 21 05:50:13 EST 2022

STATUS

editing

proposed

#142 by Peter Bala at Sun Feb 20 12:27:50 EST 2022

FORMULA

From Peter Bala, Feb 20 2022: (Start)

The o.g.f. A(x) satisfies the differential equation

(-256*x^3 + 27*x^2)*A(x)''' + (-1152*x^2 + 54*x)*A(x)'' + (-816*x + 6)*A(x)' - 24*A(x) = 0 with A(0) = 1, A'(0) = 4 and A''(0) = 56.

Algebraic equation: (1 - A(x))*(1 + 3*A(x))^3 + 256*x*A(x)^4 = 0.

Sum_{n >= 1} a(n)*( x*(3*x + 4)^3/(256*(1 + x)^4) )^n = x. (End)

CROSSREFS

binomial(k*n,n): A000984 (k = 2), A005809 (k = 3), A001449 (k = 5), A004355 (k = 6), A004368 (k = 7), A004381 (k = 8), A169958 - A169961 (k = 9 thru 12).

STATUS

approved

editing