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URL: https://oeis.org/A252354

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A252354
Number of Motzkin paths of length n with no level steps at height 2.
3
1, 1, 2, 4, 9, 20, 46, 106, 248, 584, 1389, 3329, 8047, 19607, 48167, 119287, 297829, 749632, 1902044, 4864553, 12538933, 32568528, 85224251, 224618900, 596106393, 1592429464, 4280667705, 11575188106, 31474407317, 86029586086, 236292044931, 651952466845
OFFSET
0,3
LINKS
FORMULA
a(n) = a(n-1) + Sum_{j=0..n-2} A217312(j)*a(n-j).
G.f: 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) / (32*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015
Conjecture: (-n+3)*a(n) +3*(2*n-7)*a(n-1) +(-7*n+24)*a(n-2) +2*(-7*n+36)*a(n-3) +2*(11*n-51)*a(n-4) +3*(3*n-23)*a(n-5) +(-10*n+63)*a(n-6) +3*(n-6)*a(n-7)=0. - R. J. Mathar, Sep 24 2016
MATHEMATICA
CoefficientList[Series[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 21 2015 *)
PROG
(PARI) x='x + O('x^50); Vec(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, Feb 14 2017
CROSSREFS
Sequence in context: A206119 A085748 A317097 * A052806 A329672 A218552
KEYWORD
nonn
AUTHOR
STATUS
approved