OFFSET
4,2
COMMENTS
Number of standard tableaux of shape (n+5,n-4). - Emeric Deutsch, May 30 2004
a(n) is the number of North-East paths from (0,0) to (n,n) that cross the diagonal y = x horizontally exactly twice. By symmetry, it is also the number of North-East paths from (0,0) to (n,n) that cross the diagonal y = x vertically exactly twice. Details can be found in Section 3.3 in Pan and Remmel's link. - Ran Pan, Feb 02 2016
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Robert Israel, Table of n, a(n) for n = 4..1650
Daniel Birmajer, Juan B. Gil and Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 9.
Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., Vol. 14, No. 5 (1976), pp. 395-405.
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
FORMULA
G.f.: x^4*C(x)^10, 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>=9, a(n-5)=(-1)^(n-9)*coeff(charpoly(A,x),x^9). [Milan Janjic, Jul 08 2010]
a(n) = A214292(2*n,n-5) for n > 4. - Reinhard Zumkeller, Jul 12 2012
From Robert Israel, Feb 02 2016: (Start)
D-finite with recurrence a(n+1) = 2*(n+1)*(2n+3)/((n+7)*(n-3)) * a(n).
a(n) ~ 20 * 4^n/sqrt(Pi*n^3). (End)
E.g.f.: 5*BesselI(5,2*x)*exp(2*x)/x. - Ilya Gutkovskiy, Jan 23 2017
From Amiram Eldar, Jan 02 2022: (Start)
Sum_{n>=4} 1/a(n) = 34*Pi/(45*sqrt(3)) - 44/175.
Sum_{n>=4} (-1)^n/a(n) = 53004*log(phi)/(125*sqrt(5)) - 79048/875, where phi is the golden ratio (A001622). (End)
MAPLE
seq(10*binomial(2*n+1, n-4)/(n+6), n=4..50); # Robert Israel, Feb 02 2016
MATHEMATICA
Table[10 Binomial[2 n + 1, n - 4]/(n + 6), {n, 4, 28}] (* Michael De Vlieger, Feb 03 2016 *)
PROG
(PARI) a(n) = 10*binomial(2*n+1, n-4)/(n+6); \\ Michel Marcus, Feb 02 2016
(Magma) [10*Binomial(2*n+1, n-4)/(n+6): n in [4..35]]; // Vincenzo Librandi, Feb 03 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved