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A257515 Number of 3-generalized 2-Motzkin paths of length n with no level steps H=(3,0) at odd level. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
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
EXAMPLE
For n=6 we have 9 paths: UDUDUD, H3H3 (4 options), UUDUDD, UUUDDD, UDUUDD and UUDDUD, where H3=(3,0).
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 *)
PROG
(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 */
(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
CROSSREFS
Sequence in context: A199499 A351351 A160126 * A105152 A066346 A175390
KEYWORD
nonn
AUTHOR
EXTENSIONS
Terms a(31) and beyond from Andrew Howroyd, May 01 2020
STATUS
approved

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)