%I #18 Feb 07 2017 02:48:48

%S 1,2,5,15,48,160,548,1914,6785,24335,88109,321521,1181039,4362855,

%T 16195747,60379623,225955264,848432824,3195394520,12067450014,

%U 45685766306,173350890788,659126407978,2510942564760,9582235262428,36627111558850,140214938146148

%N Number of North-East lattice paths that do not bounce off the diagonal y = x to the right.

%C This sequence is related to paired pattern P_2 in Pan and Remmel's link.

%H G. C. Greubel, <a href="/A268407/b268407.txt">Table of n, a(n) for n = 0..1000</a>

%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.: 2 (-1 + f(x) + x)/(1 - f(x) + (-5 + f(x))*x), where f(x) = sqrt(1 - 4*x).

%F a(n):= Sum_{k=0..n}((k+1)*fib(k)*binomial(2*n-k,n-k))/(n+1) + C(n), where fib(n) - Fibonacci numbers, C(n) - Catalan numbers. - _Vladimir Kruchinin_, Feb 27 2016

%F a(n) ~ 13*4^n/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Feb 27 2016

%t CoefficientList[Series[2 (-1 + Sqrt[1 - 4 x] + x) / (1 - Sqrt[1 - 4 x] + (-5 + Sqrt[1 - 4 x]) x), {x, 0, 33}], x] (* _Vincenzo Librandi_, Feb 04 2016 *)

%o (Maxima) a(n):=sum((k+1)*fib(k)*binomial(2*n-k,n-k),k,0,n)/(n+1)+binomial(2*n,n)/(n+1); /* _Vladimir Kruchinin_, Feb 27 2016 */

%Y Cf. A000045, A000108.

%K nonn

%O 0,2

%A _Ran Pan_, Feb 04 2016