A000590 a(n) = 13*binomial(2n,n-6)/(n+7). (Formerly M4908 N2104)

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

6,2

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

Cf. A000108, A001622, A214292.

Sequence in context: A282921 A023011 A022641 * A052065 A303967 A304929

Adjacent sequences: A000587 A000588 A000589 * A000591 A000592 A000593

Keyword

nonn

Author

N. J. A. Sloane

Status

approved