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a(n) = Sum_{k=0..floor(n/2)} n^(3*k) * binomial(n-k,k).
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%I #9 Jan 09 2024 08:45:47

%S 1,1,9,55,4289,47376,10358713,162592977,70065589761,1419907258279,

%T 1015035028009001,25173466118539344,26947505294538873409,

%U 790057195504021692521,1183327797361056503499225,40027334070963910087734751,79925496016112851520801796097

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

%F a(n) = [x^n] 1/(1 - x - n^3*x^2).

%F a(n) ~ n^(3*n/2) if n is even and a(n) ~ n^((3*n-1)/2)/2 if n is odd. - _Vaclav Kotesovec_, Jan 09 2024

%t Table[Hypergeometric2F1[1/2 - n/2, -n/2, -n, -4*n^3], {n, 0, 20}] (* _Vaclav Kotesovec_, Jan 09 2024 *)

%o (PARI) a(n) = sum(k=0, n\2, n^(3*k)*binomial(n-k, k));

%Y Cf. A171180, A368888.

%K nonn,easy

%O 0,3

%A _Seiichi Manyama_, Jan 09 2024