A362206 - OEIS

%I #15 Feb 19 2024 12:06:49

%S 1,1,4,25,181,1399,11212,91936,765805,6452449,54841438,469306102,

%T 4038193870,34903997029,302828905471,2635745917759,23003622046900,

%U 201241080558652,1764149626139119,15493365042402772,136288275628625410,1200600389345625754

%N Expansion of 1/(1 - x/(1-9*x)^(1/3)).

%F a(n) = Sum_{k=0..n} (-9)^k * binomial(-(n-k)/3,k) = Sum_{k=0..n} 9^(n-k) * binomial(n-2*k/3-1,n-k).

%F a(n) ~ (1-9*r)^(1/3) * (6 - 18*r + r^2) / (109 * r^n), where r = 0.1109593191262346... is the root of the equation r*(9 + r^2) = 1. - _Vaclav Kotesovec_, Feb 19 2024

%t Table[Sum[9^(n - k)*Binomial[n - 2*k/3 - 1, n - k], {k, 0, n}], {n, 0, 25}] (* _Vaclav Kotesovec_, Feb 19 2024 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(1/(1-x/(1-9*x)^(1/3)))

%Y Cf. A026671, A362157, A362210.

%K nonn,easy

%O 0,3

%A _Seiichi Manyama_, Apr 11 2023