A109192 - OEIS

%I #17 Jul 24 2022 11:14:59

%S 1,1,2,5,13,34,91,247,678,1877,5233,14674,41349,117001,332260,946527,

%T 2703915,7743268,22223607,63909987,184121946,531318553,1535522513,

%U 4443815554,12876794147,37356832679,108494114718,315415738025

%N Number of Grand Motzkin paths of length n and having no hills (i.e., no ud's starting at level 0). (A Grand Motzkin path of length n is a path in the half-plane x >= 0, starting at (0,0), ending at (n,0) and consisting of steps u=(1,1), d=(1,-1) and h=(1,0).)

%C Column 0 of A109191.

%F G.f.: 1/(z^2 + sqrt(1 - 2*z - 3*z^2)).

%F D-finite with recurrence -9*(2 + n)*(3 + n)*a(n) + (-198 - 111*n - 15*n^2)*a(n+1) + (-78 - 102*n - 24*n^2)*a(n+2) + (-462 - 340*n - 56*n^2)*a(n+3) + (-186 - 106*n - 14*n^2)*a(n+4) + (1086 + 426*n + 42*n^2)*a(n+5) + (108 + 49*n + 5*n^2)*a(n+6) + (-432 - 139*n - 11*n^2)*a(n+7) + 2*(6 + n)*(8 + n)*a(n+8) = 0. - _Benedict W. J. Irwin_, Nov 02 2016

%e a(3)=5 because we have hhh,hdu,duh,uhd and dhu.

%p g:=1/(z^2+sqrt(1-2*z-3*z^2)): gser:=series(g,z=0,33): 1,seq(coeff(gser,z^n),n=1..31);

%t CoefficientList[Series[1/(z^2+Sqrt[1-2z-3z^2]), {z, 0, 30}],z] (* _Benedict W. J. Irwin_, Nov 02 2016 *)

%Y Cf. A109191.

%K nonn

%O 0,3

%A _Emeric Deutsch_, Jun 21 2005