A000590 a(n) = 13*binomial(2n,n-6)/(n+7).

1, 13, 104, 663, 3705, 19019, 92092, 427570, 1924065, 8454225, 36463440, 154969620, 650872404, 2707475148, 11173706960, 45812198536, 186803188858, 758201178306, 3065415516592, 12352414499425, 49634247352235, 198954083924505, 795816335698020, 3177498557750790

Offset: 6

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=6. - Herbert Kociemba, May 24 2004

Number of standard tableaux of shape (n+6,n-6). - 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 = 6..200
• Richard K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., 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_] := 13*Binomial[2*n, n-6]/(n+7); Array[a, 24, 6] (* Amiram Eldar, Sep 26 2022 *)

PARI

a(n) = 13*binomial(2*n,n-6)/(n+7); \\ Michel Marcus, Oct 16 2017

Formula

G.f.: x^6*C(x)^13, where C(x)=[1-sqrt(1-4x)]/(2x) is g.f. for the Catalan numbers (A000108). - Emeric Deutsch, May 30 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>=12, a(n-6)=(-1)^(n-12)*coeff(charpoly(A,x),x^12). - Milan Janjic, Jul 08 2010

a(n) = A214292(2*n-1,n-7) for n > 6. - Reinhard Zumkeller, Jul 12 2012

-(n+7)*(n-6)*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>=6} 1/a(n) = 16777/5460 - 128*Pi/(117*sqrt(3)).

Sum_{n>=6} (-1)^n/a(n) = 787536*log(phi)/(325*sqrt(5)) - 14210999/27300, where phi is the golden ratio (A001622). (End)