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a(n) = Sum_{k=0..n} binomial(2*n, n-k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).
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%I #6 Aug 02 2022 04:13:06

%S 1,3,11,43,172,695,2823,11501,46940,191791,784148,3207196,13119733,

%T 53670793,219545353,897957702,3672093558,15013596535,61370565546,

%U 250803861369,1024716136043,4185683293934,17093143284723,69786349712519,284847779542644,1162385753008079

%N a(n) = Sum_{k=0..n} binomial(2*n, n-k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).

%F a(n) ~ 2^(2*n - 1/2) * exp(3^(1/3) * Pi^(4/3) * n^(1/3) / 2^(8/3)) / sqrt(3*Pi*n).

%t Table[Sum[PartitionsQ[k]*Binomial[2*n, n-k], {k, 0, n}], {n, 0, 30}]

%t nmax = 30; CoefficientList[Series[Sum[PartitionsQ[k]*((1-2*x-Sqrt[1-4*x])/(2*x))^k / Sqrt[1-4*x], {k, 0, nmax}], {x, 0, nmax}], x]

%Y Cf. A000009, A032443, A266232, A307496, A356268, A356280.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Aug 01 2022