A005810 a(n) = binomial(4n,n). (Formerly M3625)

1, 4, 28, 220, 1820, 15504, 134596, 1184040, 10518300, 94143280, 847660528, 7669339132, 69668534468, 635013559600, 5804731963800, 53194089192720, 488526937079580, 4495151581425648, 41432089765583440, 382460951663844400

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 congruence 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.

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.

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (terms 0..100 from T. D. Noe, terms 101..213 from Muniru A Asiru)

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].

Tom Copeland, Discriminating Deltas, Depressed Equations, and Generalized Catalan Numbers, 2012, pp. 5-6.

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

Ruppi Rana, Title? [Broken link]

Formula

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

D-finite with recurrence: 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)

a(n) = hypergeom([-3*n, -1*n], [1], 1). - Peter Luschny, Mar 19 2018

RHS of the identity Sum_{k = 0..2*n} (-1)^(n+k)*binomial(4*n, k)* binomial(4*n, 2*n-k) = binomial(4*n,n). - Peter Bala, Oct 07 2021

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)

Example

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

Maple

seq(binomial(4*n, n), n=0..20); # Muniru A Asiru, Mar 19 2018

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
(PARI) a(n) = binomial(4*n, n); \\ Altug Alkan, Mar 19 2018
(GAP) List([0..20], n->Binomial(4*n, n)); # Muniru A Asiru, Mar 19 2018
(Python)
from math import comb
def A005810(n): return comb(n<<2, n) # Chai Wah Wu, Aug 01 2023

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).

Cf. A007318, A182400, A262261.

Sequence in context: A026020 A243116 A026033 * A371755 A121203 A368234

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