A257515 Number of 3-generalized 2-Motzkin paths of length n with no level steps H=(3,0) at odd level.
1, 0, 1, 2, 2, 4, 9, 12, 26, 48, 90, 172, 348, 664, 1349, 2680, 5438, 10976, 22510, 45900, 94700, 195032, 404442, 838824, 1748308, 3646368, 7632628, 15994232, 33606168, 70699504, 149050669, 314625264, 665280246, 1408436672, 2986069782, 6337988876
Offset: 0
Keywords
Examples
For n=6 we have 9 paths: UDUDUD, H3H3 (4 options), UUDUDD, UUUDDD, UDUUDD and UUDDUD, where H3=(3,0).
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
CoefficientList[Series[(1-2*x^3-Sqrt[(1-2x^3)*(1-4*x^2-2*x^3)])/(2*x^2*(1-2*x^3)), {x, 0, 30}], x] (* Vaclav Kotesovec, Apr 28 2015 *) -
Maxima
a(n):=sum((binomial(2*m,m)/(m+1)*(if mod(n+m,3)=0 then 2^((n-2*m)/3)* binomial((m+n)/3,m) else 0)),m,0,n); /* Vladimir Kruchinin, Mar 07 2016 */
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PARI
seq(n)={Vec((1-2*x^3-sqrt((1-2*x^3)*(1-4*x^2-2*x^3) + O(x^(3+n))))/(2*x^2*(1-2*x^3)))} \\ Andrew Howroyd, May 01 2020
Formula
G.f.: (1-2*x^3-sqrt((1-2x^3)*(1-4*x^2-2*x^3)))/(2*x^2*(1-2*x^3)).
Conjecture: (n+2)*a(n) +(n+1)*a(n-1) +(n+4)*a(n-2) +4*(-2*n+3)*a(n-3) +4*(-6*n+17)*a(n-4) +4*(-3*n+10)*a(n-5) +4*(3*n-11)*a(n-6) +4*(11*n-50)*a(n-7) +20*(n-6)*a(n-8)=0. - R. J. Mathar, Jun 07 2016
Extensions
Terms a(31) and beyond from Andrew Howroyd, May 01 2020