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A268329
Expansion of (1 - sqrt(1 - 4*x))^5/16.
0
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
5,1
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
Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 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
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 *)
CROSSREFS
Sequence in context: A320526 A193519 A374298 * A223095 A052978 A351511
KEYWORD
nonn
AUTHOR
Ran Pan, Feb 01 2016
STATUS
approved