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a(n) = Sum_{k=0..n} binomial(3*k, k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).
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%I #5 Aug 02 2022 05:51:37

%S 1,4,19,187,1177,10186,84442,665842,5078668,42573268,343023418,

%T 2665464058,21440629558,167644287550,1330569327310,10641989818078,

%U 82797155054782,644097780350332,5102709814966162,39499844158337962,307777892529889642,2406854983109480302

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

%F a(n) ~ 3^(3*n + 13/4) * exp(Pi*sqrt(n/3)) / (23 * sqrt(Pi) * n^(5/4) * 2^(2*n+3)).

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

%Y Cf. A000009, A188675, A356270, A356286.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Aug 01 2022