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%I #19 Sep 08 2022 08:46:24
%S 1,9,43,155,511,1442,4131,10323,28171,63987,171667,369395,957958,
%T 2047694,5078963,10671529,26542339,53522031,132273403,268623854,
%U 647842889,1266118858,3197923083,6058756355,14581380971,29480406552,68634048862,131847974143,323289015466,611887749996
%N a(n) = Sum_{k=1..n} binomial(n,k) * sigma(k) * sigma(n - k + 1), where sigma = A000203.
%F E.g.f.: (1/2) * d/dx (Sum_{k>=1} sigma(k) * x^k / k!)^2.
%t Table[Sum[Binomial[n, k] DivisorSigma[1, k] DivisorSigma[1, n - k + 1], {k, 1, n}], {n, 1, 30}]
%t nmax = 30; CoefficientList[Series[(1/2) D[Sum[DivisorSigma[1, k] x^k/k!, {k, 1, nmax}]^2, x], {x, 0, nmax}], x] Range[0, nmax]! // Rest
%o (Magma) [&+[Binomial(n,k)*DivisorSigma(1,k)*DivisorSigma(1,n-k+1):k in [1..n]]:n in [1..30]]; // _Marius A. Burtea_, Dec 03 2019
%o (PARI) a(n) = sum(k=1, n, binomial(n,k)*sigma(k)*sigma(n-k+1)); \\ _Michel Marcus_, Dec 05 2019
%Y Cf. A000203, A000385, A007318, A185003, A328681.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Dec 03 2019