A060150 a(0) = 1; for n > 0, binomial(2n-1, n-1)^2.

1, 1, 9, 100, 1225, 15876, 213444, 2944656, 41409225, 590976100, 8533694884, 124408576656, 1828114918084, 27043120090000, 402335398890000, 6015361252737600, 90324408810638025, 1361429497505672100, 20589520178326522500, 312321918272897610000

Offset: 0

Author

N. J. A. Sloane, Apr 10 2001

Keywords

• nonn
• easy

Comments

Number of square lattice walks that start at (0,0) and end at (1,0) after 2n-1 steps, free to pass through (1,0) at intermediate steps. - Steven Finch, Dec 20 2001

Number of paths of length n connecting two neighboring nodes in optimal chordal graph of degree 4, G(2*d(G)^2+2*d(G)+1,2d(G)+1), of diameter d(G). - B. Dubalski (dubalski(AT)atr.bydgoszcz.pl), Feb 05 2002

a(n) is the number of ways to place n red balls and n blue balls into n distinguishable boxes with no restrictions on the number of balls put in a box. - Geoffrey Critzer, Jul 08 2013

The number of square lattice walks of n steps that start at the origin and end at (k,0) is zero if n-k is odd and [binomial(n,(n-k)/2)]^2 if n-k is even. - R. J. Mathar, Sep 28 2020

References

• R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 1994 Addison-Wesley company, Inc.
• A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (5.1.29.2)
• K. A. Ross and C. R. B. Wright, Discrete Mathematics, 1992 Prentice Hall Inc.

Links

• Harry J. Smith, Table of n, a(n) for n = 0..200
• R. Bacher, Meander algebras

Crossrefs

Cf. A002894, A135389, A337900, A337901, A337902.

Programs

Maple

seq(coeff(series(EllipticK(4*sqrt(x))/(2*Pi) + 3/4, x=0, n+1), x, n), n=0..30);  # Mark van Hoeij, Apr 30 2013

Mathematica

Table[Binomial[2n-1,n]^2,{n,0,19}] (* Geoffrey Critzer, Jul 08 2013 *)

PARI

a(n)=if(n<2, 1, binomial(2*n-1,n-1)^2)

PARI

for (n=0, 200, if (n==0, a=1, a=binomial(2*n - 1, n - 1)^2); write("b060150.txt", n, " ", a)) \\ Harry J. Smith, Jul 02 2009

Formula

a(n) = A088218(n)^2.

a(n) = A002894(n)/4 for n>0.

G.f.: 1 + (1/AGM(1, sqrt(1-16*x))-1)/4. - Michael Somos, Dec 12 2002

G.f. = 1 + (K(16x)-1)/4 = 1 + Sum_{k>0} q^k/(1+q^(2k)) where K(16x) is the complete Elliptic integral of the first kind at 16x=k^2 and q is the nome. - Michael Somos, May 09 2005

G.f.: 1 + x*3F2((1, 3/2, 3/2); (2, 2))(16*x). - Olivier Gérard, Feb 16 2011

E.g.f.: Sum_{n>0} a(n)*x^(2n-1)/(2n-1)! = BesselI(0, 2x)*BesselI(1, 2x) . - Michael Somos, Jun 22 2005

D-finite with recurrence n^2*a(n) -4*(2*n-1)^2*a(n-1)=0. - R. J. Mathar, Jul 26 2014

From Seiichi Manyama, Oct 19 2016: (Start)

Let the number of multisets of length k on n symbols be denoted by ((n, k)) = binomial(n+k-1, k).

a(n) = (Sum_{0 <= k <= n} binomial(n, k)^2 * ((2*n, n - k)))/3 for n > 0. (End)

a(n) ~ 4^(2*n-1)/(Pi*n). - Ilya Gutkovskiy, Oct 19 2016

For n >= 1, a(n) = 1/n * Sum_{k = 0..n-1} (n + 2*k)*binomial(n+k-1, k)^2 = ( 1/(4*n) * Sum_{k = 0..n} (n + 2*k)*binomial(-n+k-1, k)^2 )^2. - Peter Bala, Nov 02 2024