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a(n) = Sum_{k=0..n} binomial(3*n, k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).
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%I #5 Aug 02 2022 05:51:25

%S 1,4,22,214,1509,12770,107884,874365,6834843,56722759,463069914,

%T 3666488610,29512199193,233492075573,1858649112464,14890457067926,

%U 117154630898329,917101099859767,7257072314543086,56653800922475280,442687465112658972,3467083846726752495

%N a(n) = Sum_{k=0..n} binomial(3*n, k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).

%F a(n) ~ 3^(3*n + 1/4) * exp(Pi*sqrt(n/3)) / (sqrt(Pi) * n^(5/4) * 2^(2*n + 2)).

%t Table[Sum[Binomial[3*n, k] * PartitionsQ[k], {k, 0, n}], {n, 0, 30}]

%Y Cf. A000009, A188675, A356268, A356284.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Aug 01 2022