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A005810 a(n) = binomial(4n,n).
(Formerly M3625)
24
1, 4, 28, 220, 1820, 15504, 134596, 1184040, 10518300, 94143280, 847660528, 7669339132, 69668534468, 635013559600, 5804731963800, 53194089192720, 488526937079580, 4495151581425648, 41432089765583440, 382460951663844400 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Start off with 0 balls in a box. Find the number of ways you can throw 3 balls back out. Then continue to throw 4 balls into the box after each stage. (I.e., the first stage is 0. Then at the next stage there are 4 ways to throw 3 balls back out.) - Ruppi Rana (ruppirana007(AT)hotmail.com), Mar 03 2004

Central coefficients of A094527. - Paul Barry, Mar 08 2011

This is the case m = 2n in Catalan's formula (2m)!*(2n)!/(m!*(m+n)!*n!) - see Umberto Scarpis in References. - Bruno Berselli, Apr 27 2012

A generating function in terms of a (labyrinthine) solution to a depressed quartic equation is given in the Copeland link for signed A005810. - Tom Copeland, Oct 10 2012

Conjecture: a(n) == 4 (mod n^3) iff n is prime. - Gary Detlefs, Apr 03 2013

For prime p, the supercongruence a(p) = binomial(4*p,p) = 4 (mod p^3)  is a known generalization of Wolstenholme's theorem. See Mestrovic, Section 6, equation 35. - Peter Bala, Dec 28 2014

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Umberto Scarpis, Sui numeri primi e sui problemi dell'analisi indeterminata in Questioni riguardanti le matematiche elementari, Nicola Zanichelli Editore (1924-1927, third Edition), page 11.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..100

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Ruppi Rana, Title? [Broken link]

R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

FORMULA

a(n) is asymptotic to c*sqrt(n)*(256/27)^n with c = 0.46... - Benoit Cloitre, Jan 26 2003

a(n) is asymptotic to c*(256/27)^n/sqrt(n) with c = sqrt(2 / (3 Pi)) = 0.460658865961780639... - Benoit Cloitre, Jan 26 2003; corrected by Charles R Greathouse IV, Dec 14 2006

a(n) = Sum_{k=0..2n} binomial(2n,k)*binomial(2n,k-n)}. - Paul Barry, Mar 08 2011

G.f.: g/(4-3*g) where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011

3*n*(3*n-1)*(3*n-2)*a(n) - 8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1) = 0. - R. J. Mathar, Dec 02 2012

a(n) = binomial(4*n,n-1)*(3*n+1)/n. - Gary Detlefs, Apr 03 2013

a(n) = C(4*n-1,n-1)*C(16*n^2,2)/(3*n*C(4*n+1,3)), n>0. - Gary Detlefs, Jan 02 2014

a(n) = Sum_{i,j,k=0..n} binomial(n,i)*binomial(n,j)*binomial(n,k)*binomial(n,i+j+k). - Peter Bala, Dec 28 2014

a(n) = GegenbauerC(n, -2*n, -1). - Peter Luschny, May 07 2016

From Ilya Gutkovskiy, Nov 22 2016: (Start)

O.g.f.: 3F2(1/4,1/2,3/4; 1/3,2/3; 256*x/27).

E.g.f.: 3F3(1/4,1/2,3/4; 1/3,2/3,1; 256*x/27). (End)

EXAMPLE

G.f. = 1 + 4*x + 28*x^2 + 220*x^3 + 1820*x^4 + 15504*x^5 + 134596*x^6 + ...

MATHEMATICA

Table[Binomial[4n, n], {n, 0, 19}] (* Geoffrey Critzer, Sep 15 2013 *)

PROG

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

(Haskell)

a005810 n = a007318 (4*n) n  -- Reinhard Zumkeller, Mar 04 2012

CROSSREFS

Cf. A007318, A182400, A262261.

Sequence in context: A026020 A243116 A026033 * A121203 A192620 A180708

Adjacent sequences:  A005807 A005808 A005809 * A005811 A005812 A005813

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Henry Bottomley, Oct 06 2000

Corrected by T. D. Noe, Jan 16 2007

STATUS

approved

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Last modified March 28 02:14 EDT 2017. Contains 284182 sequences.