A268329 Expansion of (1 - sqrt(1 - 4*x))^5/16.

2, 10, 40, 150, 550, 2002, 7280, 26520, 96900, 355300, 1307504, 4828850, 17895150, 66533250, 248124000, 927983760, 3479939100, 13082337900, 49295766000, 186156379500, 704415740028, 2670587146260, 10142836030240, 38586876202000, 147029304149000

Offset: 5

Author

Ran Pan, Feb 01 2016

Keywords

• nonn

Comments

a(n) is the number of North-East paths from (0,0) to (n,n) that cross the diagonal vertically exactly once and horizontally exactly twice, and bounce off the diagonal to the right once but not to the left. Details about this sequence can be found in Section 4.5 in Pan and Remmel's link. - Ran Pan, Feb 01 2016

Links

• Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Crossrefs

Cf. A000344, A000984.

Programs

Mathematica

Table[10 Binomial[2 n - 6, n - 5]/n, {n, 5, 29}] (* or *)
Table[SeriesCoefficient[(1 - Sqrt[1 - 4 x])^5/16, {x, 0, n}], {n, 5, 29}] (* Michael De Vlieger, Feb 17 2016 *)

Formula

G.f.: (1 - sqrt(1 - 4*x))^5/16.

a(n) = 10 * binomial(2n-6,n-5)/n.

a(n) = 2*A000344(n-3). - R. J. Mathar, Feb 17 2016

D-finite with recurrence: n*(n-5)*a(n) -2*(n-3)*(2*n-7)*a(n-1)=0. - R. J. Mathar, Feb 17 2016