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%I #22 Jan 30 2020 21:29:17
%S 2,10,40,150,550,2002,7280,26520,96900,355300,1307504,4828850,
%T 17895150,66533250,248124000,927983760,3479939100,13082337900,
%U 49295766000,186156379500,704415740028,2670587146260,10142836030240,38586876202000,147029304149000
%N Expansion of (1 - sqrt(1 - 4*x))^5/16.
%C 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
%H Ran Pan, Jeffrey B. Remmel, <a href="http://arxiv.org/abs/1601.07988">Paired patterns in lattice paths</a>, arXiv:1601.07988 [math.CO], 2016.
%F G.f.: (1 - sqrt(1 - 4*x))^5/16.
%F a(n) = 10 * binomial(2n-6,n-5)/n.
%F a(n) = 2*A000344(n-3). - _R. J. Mathar_, Feb 17 2016
%F 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
%t Table[10 Binomial[2 n - 6, n - 5]/n, {n, 5, 29}] (* or *)
%t Table[SeriesCoefficient[(1 - Sqrt[1 - 4 x])^5/16, {x, 0, n}], {n, 5, 29}] (* _Michael De Vlieger_, Feb 17 2016 *)
%Y Cf. A000344, A000984.
%K nonn
%O 5,1
%A _Ran Pan_, Feb 01 2016
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