OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Beáta Bényi, Toufik Mansour, and José L. Ramírez, Set partitions and non-crossing partitions with l-neighbors and l-isolated elements, Australasian J. Comb. (2022) Vol. 84, No. 2, 325-340.
Emeric Deutsch, Ordered trees with prescribed root degrees, node degrees and branch lengths, Discrete Math., 282, 2004, 89-94. See p. 93.
FORMULA
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
MAPLE
beta := proc(n, k)
coeftayl((x+x^2+x^3)^k, x=0, n) ;
end proc:
A239106 := proc(n)
add( A001006(k-1)*beta(n, k), k=ceil(n/3)..n) ;
end proc: # R. J. Mathar, Apr 02 2014
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Mar 26 2014
STATUS
approved