A268407 Number of North-East lattice paths that do not bounce off the diagonal y = x to the right.

1, 2, 5, 15, 48, 160, 548, 1914, 6785, 24335, 88109, 321521, 1181039, 4362855, 16195747, 60379623, 225955264, 848432824, 3195394520, 12067450014, 45685766306, 173350890788, 659126407978, 2510942564760, 9582235262428, 36627111558850, 140214938146148

Offset: 0

Author

Ran Pan, Feb 04 2016

Keywords

• nonn

Comments

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Crossrefs

Cf. A000045, A000108.

Programs

Mathematica

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 *)

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 */

Formula

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

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

a(n) ~ 13*4^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 27 2016