cp's OEIS Frontend

A268400 Number of North-East lattice paths from (0,0) to (n,n) that bounce off the diagonal y = x to the right exactly twice.

%I A268400 #23 Jul 24 2022 14:37:30
%S A268400 1,5,23,99,413,1691,6842,27464,109631,435887,1728018,6835668,26996393,
%T A268400 106486529,419639903,1652533719,6504159137,25589302163,100646529977,
%U A268400 395775842389,1556107102849,6117771240251,24050813530815,94550689834203,371715533473021,1461430355605367,5746128800657639,22594839306797223
%N A268400 Number of North-East lattice paths from (0,0) to (n,n) that bounce off the diagonal y = x to the right exactly twice.
%C A268400 This sequence is related to paired pattern P_2 in Pan and Remmel's link.
%H A268400 G. C. Greubel, <a href="/A268400/b268400.txt">Table of n, a(n) for n = 3..1000</a>
%H A268400 Ran Pan and Jeffrey B. Remmel, <a href="http://arxiv.org/abs/1601.07988">Paired patterns in lattice paths</a>, arXiv:1601.07988 [math.CO], 2016.
%F A268400 G.f.: -((-1 + f(x))^3*x*(-1 + f(x) + 2*x))/(2*(1 - f(x) + (-5 + f(x))*x)^3), where f(x) = sqrt(1 - 4*x).
%F A268400 a(n) = ((Sum_{m=1..n-3}((m+2)*(Sum_{k=0..m/2}((m^2+(3-2*k)*m+k^2-3*k+2)*binomial(m-k,k)))*binomial(2*n-m-5,n-m-3)))+2*binomial(2*n-4,n-2))/(2*n-2), n>2. - _Vladimir Kruchinin_, Feb 28 2016
%F A268400 a(n) ~ 7*2^(2*n+2)/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Feb 28 2016
%F A268400 D-finite with recurrence (n-1)*(9*n-56)*a(n) +(-31*n^2+203*n-16)*a(n-1) +(-193*n^2+1371*n-2138)*a(n-2) +3*(217*n^2-1497*n+2274)*a(n-3) +2*(41*n-94)*(2*n-5)*a(n-4)=0. - _R. J. Mathar_, Jul 24 2022
%t A268400 Rest[Rest[Rest[CoefficientList[Series[-((-1 + Sqrt[1 - 4 x])^3 x (-1 + Sqrt[1-4 x] + 2 x))/(2 (1 - Sqrt[1 - 4 x] + (-5 + Sqrt[1 - 4 x]) x)^3), {x, 0, 40}], x]]]] (* _Vincenzo Librandi_, Feb 28 2016 *)
%o A268400 (Maxima)
%o A268400 a(n):=((sum((m+2)*(sum((m^2+(3-2*k)*m+k^2-3*k+2)*binomial(m-k,k),k,0,m/2)) *binomial(2*n-m-5,n-m-3),m,1,n-3))+2*binomial(2*n-4,n-2))/(2*n-2); /* _Vladimir Kruchinin_, Feb 28 2016 */
%Y A268400 Cf. A000108, A001628, A033184.
%K A268400 nonn
%O A268400 3,2
%A A268400 _Ran Pan_, Feb 03 2016