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, 561063824632584

Offset

5,1

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

Table of n, a(n) for n=5..30.

Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016-2017.

Formula

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

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

From R. J. Mathar, Feb 17 2016: (Start)

a(n) = 2*A000344(n-3).

D-finite with recurrence: n*(n-5)*a(n) - 2*(n-3)*(2*n-7)*a(n-1) = 0. (End)

E.g.f.: (15*(2 - 10*x + 5*x^2) + 2*exp(2*x)*(16*x^3 - 128*x^2 + 105*x - 15)*BesselI(0, 2*x) - exp(2*x)*x*(32*x^2 - 248*x + 149)*BesselI(1, 2*x))/30. - Stefano Spezia, Sep 21 2025

a(n) ~ 5 * 2^(2*n-5) / (n^(3/2) * sqrt(Pi)). - Amiram Eldar, Oct 19 2025

Mathematica

Table[10 Binomial[2 n - 6, n - 5]/n, {n, 5, 30}] (* or *)
Table[SeriesCoefficient[(1 - Sqrt[1 - 4 x])^5/16, {x, 0, n}], {n, 5, 30}] (* Michael De Vlieger, Feb 17 2016 *)

Crossrefs

Cf. A000344, A000984.

Sequence in context: A320526 A193519 A374298 * A390683 A223095 A052978

Adjacent sequences: A268326 A268327 A268328 * A268330 A268331 A268332

Keyword

nonn,easy

Author

Ran Pan, Feb 01 2016

Extensions

a(30) from Stefano Spezia, Sep 21 2025

Status

approved