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%I #4 Aug 02 2022 06:40:19
%S 1,5,31,200,1309,8627,57082,378648,2516111,16740913,111494801,
%T 743137984,4956359312,33074272702,220810039566,1474764797488,
%U 9853307017341,65853733243281,440255398634199,2944041287677060,19691951641479427,131744163990056479,881586559906575688
%N a(n) = Sum_{k=0..n} binomial(3*n, n-k) * v(k), where v(k) is the number of overpartitions of n (A015128).
%F a(n) ~ c * 3^(3*n + 1/2) / (sqrt(Pi*n) * 2^(2*n + 1)), where c = Sum_{j>=0} v(j)/2^j = 8.2559879357782500655441408494322731265270016167882303456037...
%t Table[Sum[Sum[PartitionsP[k-j]*PartitionsQ[j], {j, 0, k}] * Binomial[3*n, n-k], {k, 0, n}], {n, 0, 30}]
%Y Cf. A015128, A266497, A356280, A356281, A356282, A356283, A356289.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Aug 02 2022