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%I #6 May 28 2019 16:49:51
%S 1,2,22,116,806,4028,26876,132776,808710,4170604,23586836,120316888,
%T 673359196,3383189976,18228019512,92654842960,486382544838,
%U 2443171359820,12694346947492,63262412763896,323739858349684,1609270321201800,8117702067063368,40102791350319408
%N a(n) = 4^n * [x^n] sqrt(1-x) * Product_{k>=1} 1/(1 - x^k).
%F a(n) ~ sqrt(Pi) * exp(Pi*sqrt(2*n/3)) * 2^(2*n - 9/4) / (3^(3/4) * n^(5/4)).
%F a(n) = A325947(n) - 4*A325947(n-1).
%t nmax = 30; CoefficientList[Series[(1-x)^(1/2) * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] * 4^Range[0, nmax]
%t Table[4^n*(PartitionsP[n] - Sum[PartitionsP[n-k] * CatalanNumber[k-1]/2^(2*k - 1), {k, 1, n}]), {n, 0, 30}]
%Y Cf. A000041, A325947, A325949.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, May 28 2019
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