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A239106
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Ordered trees with branches of length at most 3.
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1
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1, 1, 2, 5, 13, 39, 121, 388, 1277, 4288, 14630, 50575, 176762, 623563, 2217379, 7939821, 28603591, 103600632, 377033451, 1378023887, 5056021292, 18615654196, 68758804039, 254706453524, 946038872000, 3522439937992, 13145057553230, 49157901220299
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = Sum_{k= ceiling(n/3)..n} trinomial(k,3;n-k) A001006(k-1).
G.f.: (1+x+x^2+x^3-sqrt(-3*x^6-6*x^5-9*x^4-8*x^3-5*x^2-2*x+1))/ (2*x^3+2*x^2+2*x). - Vladimir Kruchinin, Mar 03 2016
Recurrence: (n+1)*a(n) = (n-2)*a(n-1) + 6*(n-2)*a(n-2) + 3*(5*n - 13)*a(n-3) + (22*n - 83)*a(n-4) + (23*n - 112)*a(n-5) + 18*(n-6)*a(n-6) + 9*(n-7)*a(n-7) + 3*(n-8)*a(n-8). - Vaclav Kotesovec, Mar 03 2016
a(n) ~ sqrt(3*(3 - 1/(3-2*sqrt(2))^(1/3) - (3-2*sqrt(2))^(1/3))/2) * ((108-54*sqrt(2))^(1/3)/3 + (4+2*sqrt(2))^(1/3) + 1)^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 03 2016
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MAPLE
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beta := proc(n, k)
coeftayl((x+x^2+x^3)^k, x=0, n) ;
end proc:
add( A001006(k-1)*beta(n, k), k=ceil(n/3)..n) ;
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MATHEMATICA
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CoefficientList[Series[(1 + x + x^2 + x^3 - Sqrt[-3*x^6 - 6*x^5 - 9*x^4 - 8*x^3 - 5*x^2 - 2*x + 1])/(2*x^3 + 2*x^2 + 2*x), {x, 0, 50}], x] (* G. C. Greubel, Apr 05 2017 *)
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PROG
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(PARI) x='x+O('x^50); Vec((1+x+x^2+x^3-sqrt(-3*x^6-6*x^5-9*x^4-8*x^3-5*x^2-2*x+1))/ (2*x^3+2*x^2+2*x)) \\ G. C. Greubel, Apr 05 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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