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a(n) = Sum_{k=0..n} binomial(3*n, 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 05:51:12

%S 1,4,22,131,807,5066,32188,206242,1329733,8614685,56024538,365491218,

%T 2390613557,15671221522,102925324569,677110860689,4460956827127,

%U 29427611146335,194348311824025,1284856925961827,8502252246841668,56309476194587377,373220349572126265

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

%F a(n) ~ c * 3^(3*n + 1/2) / (sqrt(Pi*n) * 2^(2*n + 1)), where c = Sum_{j>=0} q(j)/2^j = A079555 = 2.384231029031371724149899288678...

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

%Y Cf. A000009, A188675, A356281, A356282.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Aug 01 2022