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A386958 a(n) = Sum_{k=0..n} 8^k * binomial(k-2/3,k) * binomial(2*n+1/3,n-k).

%I A386958 #19 Sep 03 2025 14:43:53
%S A386958 1,5,33,248,2020,17325,153699,1395084,12868839,120127865,1131633217,
%T A386958 10737438816,102480890512,982880111192,9465545374920,91479218990688,
%U A386958 886803360846876,8619761335490460,83982810424366860,819973263265010400,8020986875021209320
%N A386958 a(n) = Sum_{k=0..n} 8^k * binomial(k-2/3,k) * binomial(2*n+1/3,n-k).
%H A386958 Vincenzo Librandi, <a href="/A386958/b386958.txt">Table of n, a(n) for n = 0..800</a>
%F A386958 a(n) = [x^n] 1/((1-9*x)^(1/3) * (1-x)^(n+1)).
%F A386958 a(n) = Sum_{k=0..n} 9^k * (-8)^(n-k) * binomial(2*n+1/3,k) * binomial(2*n-k,n-k).
%F A386958 a(n) = Sum_{k=0..n} 9^k * binomial(k-2/3,k) * binomial(2*n-k,n-k).
%F A386958 G.f.: 1/( sqrt(1-4*x) * ((9*sqrt(1-4*x)-7)/2)^(1/3) ).
%t A386958 Table[Sum[8^k*Binomial[k-2/3,k]*Binomial[2*n+1/3, n-k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Sep 03 2025 *)
%o A386958 (PARI) a(n) = sum(k=0, n, 8^k*binomial(k-2/3, k)*binomial(2*n+1/3, n-k));
%Y A386958 Cf. A386957.
%K A386958 nonn,changed
%O A386958 0,2
%A A386958 _Seiichi Manyama_, Aug 11 2025