A257386 Number of Motzkin paths of length n with no level steps at height 3.
1, 1, 2, 4, 9, 21, 51, 126, 316, 799, 2034, 5202, 13357, 34407, 88888, 230237, 597829, 1555962, 4058944, 10612102, 27807135, 73025751, 192204957, 507025163, 1340545113, 3552492126
Offset: 0
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
CoefficientList[Series[1/(1-x-x^2*(1/(1-x-x^2*(1/(1-x-x^2*(1+x-Sqrt[1-2*x-3*x^2])/(2*x*(1+x))))))), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 24 2015 *) -
PARI
x='x+O('x^50); Vec(1/(1-x-x^2*(1/(1-x-x^2*(1/(1-x-x^2*(1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x)))))))) \\ G. C. Greubel, Apr 08 2017
Formula
a(n) = a(n-1) + Sum_{j=0..n-2} A252354(j)*a(n-j).
G.f: 1/(1-x-x^2*(1/(1-x-x^2*(1/(1-x-x^2*R(x)))))), where R(x) is the g.f. of Riordan numbers (A005043).
a(n) ~ 3^(n+3/2)/(50*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 24 2015