A000589 a(n) = 11*binomial(2n,n-5)/(n+6).

1, 11, 77, 440, 2244, 10659, 48279, 211508, 904475, 3798795, 15737865, 64512240, 262256280, 1059111900, 4254603804, 17018415216, 67837293986, 269638992062, 1069258071970, 4232010895376, 16723268860760, 65997186039785, 260170725132045, 1024713341952300

Offset: 5

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch but do not cross the line x-y=5. - Herbert Kociemba, May 24 2004

Number of standard tableaux of shape (n+5,n-5). - Emeric Deutsch, May 30 2004

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 5..200
• Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
• Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Applied Math. 14 (1956), 405ff.
• Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math 14 (1957), 405-414. [Annotated scan of selected pages]
• John Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.

Crossrefs

Cf. A000108, A001622, A214292.

Programs

Mathematica

a[n_] := 11*Binomial[2*n, n-5]/(n+6); Array[a, 25, 5] (* Amiram Eldar, Sep 26 2022 *)

Formula

Expansion of x^5*C^11, where C = (1-(1-4*x)^(1/2))/(2*x) is the g.f. for the Catalan numbers, A000108. - Philippe Deléham, Feb 03 2004

Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=10, a(n-5)=(-1)^(n-10)*coeff(charpoly(A,x),x^10). - Milan Janjic, Jul 08 2010

a(n) = A214292(2*n-1,n-6) for n > 5. - Reinhard Zumkeller, Jul 12 2012

-(n+6)*(n-5)*a(n) + 2*n*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Jun 20 2013

From Amiram Eldar, Sep 26 2022: (Start)

Sum_{n>=5} 1/a(n) = 5993/1540 - 152*Pi/(99*sqrt(3)).

Sum_{n>=5} (-1)^(n+1)/a(n) = 210624*log(phi)/(275*sqrt(5)) - 1262077/7700, where phi is the golden ratio (A001622). (End)