A003519 a(n) = 10*C(2n+1, n-4)/(n+6).

1, 10, 65, 350, 1700, 7752, 33915, 144210, 600875, 2466750, 10015005, 40320150, 161280600, 641886000, 2544619500, 10056336264, 39645171810, 155989499540, 612815891050, 2404551645100, 9425842448792, 36921502679600, 144539291740025, 565588532895750, 2212449261033375

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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.

Crossrefs

A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.

Cf. A000108, A000245, A002057, A000344, A003517, A000588, A003518, A001392, A001622.

Programs

Magma

[10*Binomial(2*n+1, n-4)/(n+6): n in [4..35]]; // Vincenzo Librandi, Feb 03 2016

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 *)

PARI

a(n) = 10*binomial(2*n+1, n-4)/(n+6); \\ Michel Marcus, Feb 02 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)