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A386959 a(n) = Sum_{k=0..n} 8^k * binomial(k-2/3,k) * binomial(2*n-2/3,n-k).

%I A386959 #21 Sep 04 2025 08:41:03
%S A386959 1,4,27,208,1724,14952,133581,1217976,11269359,105423292,994691555,
%T A386959 9449623872,90277420688,866526247552,8350536475896,80748593332416,
%U A386959 783157950294876,7615517087165040,74225719019229060,724945200854844480,7093481177196998640
%N A386959 a(n) = Sum_{k=0..n} 8^k * binomial(k-2/3,k) * binomial(2*n-2/3,n-k).
%H A386959 Vincenzo Librandi, <a href="/A386959/b386959.txt">Table of n, a(n) for n = 0..300</a>
%F A386959 a(n) = [x^n] 1/((1-9*x)^(1/3) * (1-x)^n).
%F A386959 a(n) = Sum_{k=0..n} 9^k * (-8)^(n-k) * binomial(2*n-2/3,k) * binomial(2*n-k-1,n-k).
%F A386959 a(n) = Sum_{k=0..n} 9^k * binomial(k-2/3,k) * binomial(2*n-k-1,n-k).
%F A386959 G.f.: (1+sqrt(1-4*x))/( 2 * sqrt(1-4*x) * ((9*sqrt(1-4*x)-7)/2)^(1/3) ).
%F A386959 D-finite with recurrence 32*n*(n-1)*a(n) -4*(n-1)*(215*n-376)*a(n-1) +3*(2353*n^2-9810*n+9920)*a(n-2) -918*(3*n-7)*(6*n-17)*a(n-3)=0. - _R. J. Mathar_, Aug 19 2025
%t A386959 Table[Sum[8^k* Binomial[ k-2/3,k]*Binomial[2*n-2/3,n-k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Sep 04 2025 *)
%o A386959 (PARI) a(n) = sum(k=0, n, 8^k*binomial(k-2/3, k)*binomial(2*n-2/3, n-k));
%o A386959 (Magma) m:=30; R<x>:=LaurentSeriesRing(RationalField(), m); Coefficients(R!((1+Sqrt(1-4*x))/( 2 * Sqrt(1-4*x) * ((9*Sqrt(1-4*x)-7)/2)^(1/3) ))); // _Vincenzo Librandi_, Sep 04 2025
%Y A386959 Cf. A386956, A386960.
%K A386959 nonn,changed
%O A386959 0,2
%A A386959 _Seiichi Manyama_, Aug 11 2025