OFFSET
6,2
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.
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)
MATHEMATICA
a[n_] := 13*Binomial[2*n, n-6]/(n+7); Array[a, 24, 6] (* Amiram Eldar, Sep 26 2022 *)
PROG
(PARI) a(n) = 13*binomial(2*n, n-6)/(n+7); \\ Michel Marcus, Oct 16 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved