%I M4908 N2104 #45 Sep 26 2022 05:46:28
%S 1,13,104,663,3705,19019,92092,427570,1924065,8454225,36463440,
%T 154969620,650872404,2707475148,11173706960,45812198536,186803188858,
%U 758201178306,3065415516592,12352414499425,49634247352235,198954083924505,795816335698020,3177498557750790
%N a(n) = 13*binomial(2n,n-6)/(n+7).
%C 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
%C Number of standard tableaux of shape (n+6,n-6). - _Emeric Deutsch_, May 30 2004
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000590/b000590.txt">Table of n, a(n) for n = 6..200</a>
%H Richard K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, Sandsteps and Pascal Pyramids</a>, J. Integer Seqs., Vol. 3 (2000), Article 00.1.6.
%H Athanasios Papoulis, <a href="http://www.jstor.org/stable/43636019">A new method of inversion of the Laplace transform</a>, Quart. Applied Math. 14 (1956), 405ff.
%H Athanasios Papoulis, <a href="/A000108/a000108_8.pdf">A new method of inversion of the Laplace transform</a>, Quart. Appl. Math 14 (1957), 405-414. [Annotated scan of selected pages]
%H John Riordan, <a href="https://doi.org/10.1090/S0025-5718-1975-0366686-9">The distribution of crossings of chords joining pairs of 2n points on a circle</a>, Math. Comp., 29 (1975), 215-222.
%F 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
%F 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
%F a(n) = A214292(2*n-1,n-7) for n > 6. - _Reinhard Zumkeller_, Jul 12 2012
%F -(n+7)*(n-6)*a(n) + 2*n*(2*n-1)*a(n-1) = 0. - _R. J. Mathar_, Jun 20 2013
%F From _Amiram Eldar_, Sep 26 2022: (Start)
%F Sum_{n>=6} 1/a(n) = 16777/5460 - 128*Pi/(117*sqrt(3)).
%F Sum_{n>=6} (-1)^n/a(n) = 787536*log(phi)/(325*sqrt(5)) - 14210999/27300, where phi is the golden ratio (A001622). (End)
%t a[n_] := 13*Binomial[2*n, n-6]/(n+7); Array[a, 24, 6] (* _Amiram Eldar_, Sep 26 2022 *)
%o (PARI) a(n) = 13*binomial(2*n,n-6)/(n+7); \\ _Michel Marcus_, Oct 16 2017
%Y Cf. A000108, A001622, A214292.
%K nonn
%O 6,2
%A _N. J. A. Sloane_