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%I #12 Apr 08 2024 04:51:04
%S 1,1,3,13,51,201,834,3529,15075,65431,288278,1285263,5799470,26492103,
%T 122432628,572291385,2705760291,12937116213,62542367166,305668511259,
%U 1510080076410,7539381024297,38034307340076,193835252945487,997724306958606,5185731234177001
%N a(n) = Sum_{k=0..floor(n/3)} n^k * binomial(2*n-3*k-1,n-3*k).
%F a(n) = [x^n] 1/((1-n*x^3) * (1-x)^n).
%F a(n) ~ exp(n^(2/3) + n^(1/3)/2 + 1/3) * n^(n/3) / 3. - _Vaclav Kotesovec_, Apr 08 2024
%t Join[{1}, Table[Sum[n^k*Binomial[2*n-3*k-1,n-1], {k, 0, n/3}], {n, 1, 25}]] (* _Vaclav Kotesovec_, Apr 08 2024 *)
%o (PARI) a(n) = sum(k=0, n\3, n^k*binomial(2*n-3*k-1, n-3*k));
%Y Cf. A293574, A371836.
%Y Cf. A120305, A371827.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Apr 08 2024
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