A109192 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).)
1, 1, 2, 5, 13, 34, 91, 247, 678, 1877, 5233, 14674, 41349, 117001, 332260, 946527, 2703915, 7743268, 22223607, 63909987, 184121946, 531318553, 1535522513, 4443815554, 12876794147, 37356832679, 108494114718, 315415738025
Offset: 0
Keywords
Examples
a(3)=5 because we have hhh,hdu,duh,uhd and dhu.
Crossrefs
Cf. A109191.
Programs
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Maple
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);
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Mathematica
CoefficientList[Series[1/(z^2+Sqrt[1-2z-3z^2]), {z, 0, 30}],z] (* Benedict W. J. Irwin, Nov 02 2016 *)
Formula
G.f.: 1/(z^2 + sqrt(1 - 2*z - 3*z^2)).
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
Comments