A268315 -id:A268315 - cp's OEIS frontend

A268316 a(n) is the number of Dyck paths of length 4n and height n.

1, 1, 7, 57, 484, 4199, 36938, 328185, 2937932, 26457508, 239414383, 2175127695, 19827974412, 181266501290, 1661241473220, 15257624681145, 140400178555644, 1294141164447692, 11946771748196428, 110435320379615620, 1022108852175416720, 9470416604629933935

Offset: 0

Gheorghe Coserea, Feb 01 2016

Equivalently, a(n) is the number of rooted plane trees with 2n+1 nodes and height n.

			For n = 2 the a(2) = 7 solutions are
              /\/\/\       |
LLRLRLRR     /      \     /|\
................................
                /\        /|\
LRLLRRLR     /\/  \/\      |
................................
              /\  /\       /\
LLRRLLRR     /  \/  \     /  \
................................
              /\           /|\
LLRRLRLR     /  \/\/\     /
................................
                  /\      /|\
LRLRLLRR     /\/\/  \        \
................................
              /\/\         /\
LLRLRRLR     /    \/\     /\
................................
                /\/\       /\
LRLLRLRR     /\/    \       /\

Gheorghe Coserea, Table of n, a(n) for n = 0..1000.
Gheorghe Coserea, Solutions for n=3.
Gheorghe Coserea, Solutions for n=4.
Gheorghe Coserea, MiniZinc model for generating solutions.

Cf. A080936, A268315.

Column k=2 of A289481.

[Binomial(4*n, n)*2*(2*n+3)*(2*n^2+1)/((3*n+1)*(3*n+2)*(3*n+3)): n in [1..30]]; // Vincenzo Librandi, Feb 04 2016

Table[Binomial[4 n, n] 2 (2 n + 3) (2 n^2 + 1) / ((3 n + 1) (3 n + 2) (3 n + 3)), {n, 1, 25}] (* Vincenzo Librandi, Feb 04 2016 *)
Drop[CoefficientList[Series[-((-1 + HypergeometricPFQ[{-3/4, -1/2, -1/4}, {1/3, 2/3}, 256 x/27])/(4x)) + 4/5 x HypergeometricPFQ[{5/4, 3/2, 7/4}, {7/3, 8/3}, 256 x/27] + 8/3 x^2 HypergeometricPFQ[{9/4, 5/2, 11/4}, {10/3, 11/3}, 256x/27], {x, 0, 20}], x], 1] (* Benedict W. J. Irwin, Aug 09 2016 *)

a(n) = binomial(4*n,n) * 2*(2*n+3)*(2*n^2+1)/((3*n+1)*(3*n+2)*(3*n+3));
vector(21, i, a(i))

a(n) = T(2n,n), where T(n,k) is defined by A080936.
a(n) = binomial(4*n,n) * 2*(2*n+3)*(2*n^2+1)/((3*n+1)*(3*n+2)*(3*n+3)).
a(n) ~ K * A268315^n / sqrt(n), where K = 8/27 * sqrt(2/(3*Pi)) = 0.13649151584...
G.f.: -((3F2(-3/4, -1/2, -1/4; 1/3, 2/3; 256*x/27)-1)/(4*x)) + 4/5*x*3F2(5/4, 3/2, 7/4; 7/3, 8/3; 256*x/27) + 8/3*x^2*3F2(9/4, 5/2, 11/4; 10/3, 11/3; 256*x/27). - Benedict W. J. Irwin, Aug 09 2016
Recurrence: 3*(n+1)*(2*n + 1)*(3*n + 1)*(3*n + 2)*(2*n^2 - 4*n + 3)*a(n) = 8*(2*n - 1)*(2*n + 3)*(4*n - 3)*(4*n - 1)*(2*n^2 + 1)*a(n-1). - Vaclav Kotesovec, Aug 10 2016

Added a(0)=1, adjusted b-file - N. J. A. Sloane, Dec 22 2016

A210621 Decimal expansion of 256/81.

Original entry on oeis.org

3, 1, 6, 0, 4, 9, 3, 8, 2, 7, 1, 6, 0, 4, 9, 3, 8, 2, 7, 1, 6, 0, 4, 9, 3, 8, 2, 7, 1, 6, 0, 4, 9, 3, 8, 2, 7, 1, 6, 0, 4, 9, 3, 8, 2, 7, 1, 6, 0, 4, 9, 3, 8, 2, 7, 1, 6, 0, 4, 9, 3, 8, 2, 7, 1, 6, 0, 4, 9, 3, 8, 2, 7, 1, 6, 0, 4, 9, 3, 8, 2, 7, 1, 6, 0, 4, 9, 3, 8, 2, 7, 1, 6, 0, 4, 9, 3, 8, 2

Offset: 1

N. J. A. Sloane, Mar 24 2012

According to Maor (1994), the Rhind Papyrus asserts that a circle has the same area as a square with a side that is 8/9 the diameter of the circle. From this we can determine that 256/81 is one of the ancient Egyptian approximations of Pi. - Alonso del Arte, Jun 12 2012

			3.1604938271604938271604938271604938271604938271604938271604...

Petr Beckmann, A History of Pi, 3rd Ed., Boulder, Colorado: The Golem Press (1974): p. 12.
Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Perseus Books, 1996, p. 88.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 237.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.6 The Quest for Pi, p. 89.
Carl Theodore Heisel, Behold! The grand problem no longer unsolved: The circle squared beyond refutation, c. 1935. (proposes Pi = 3 + 13/81)
Eli Maor, e: The Story of a Number. Princeton, New Jersey: Princeton University Press (1994): 41, 47 note 1.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 48.

Dario Castellanos, The ubiquitous Pi, Math. Mag., 61 (1988), 67-98 and 148-163.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Cf. A068028, A229943, A255910, A268315.

RealDigits[256/81, 10, 100][[1]] (* Alonso del Arte, Jun 12 2012 *)

256/81. \\ Charles R Greathouse IV, Sep 13 2013

256/81 = (4/3)^4.

Equals 3*A229943 = A255910^2 = A268315/3. - Hugo Pfoertner, Jun 26 2024

Offset corrected by Rick L. Shepherd, Jan 06 2014

cp's OEIS Frontend

A268316 a(n) is the number of Dyck paths of length 4n and height n.

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