A328681 - OEIS

%I #30 Sep 08 2022 08:46:24

%S 1,6,20,55,142,322,779,1608,3894,7370,18372,33137,81512,149694,353224,

%T 641461,1570836,2684928,6642915,11795178,28133846,46768200,125433400,

%U 197654545,485749918,893864394,2066417482,3385115393,8975476976,14384181908,35478028091,61940000322

%N a(n) = Sum_{k=1..n} binomial(n,k) * tau(k) * tau(n - k + 1), where tau = A000005.

%F E.g.f.: (1/2) * d/dx (Sum_{k>=1} tau(k) * x^k / k!)^2.

%t Table[Sum[Binomial[n, k] DivisorSigma[0, k] DivisorSigma[0, n - k + 1], {k, 1, n}], {n, 1, 32}]

%t nmax = 32; CoefficientList[Series[(1/2) D[Sum[DivisorSigma[0, k] x^k/k!, {k, 1, nmax}]^2, x], {x, 0, nmax}], x] Range[0, nmax]! // Rest

%o (Magma) [&+[Binomial(n,k)*DivisorSigma(0,k)*DivisorSigma(0,n-k+1):k in [1..n]]:n in [1..32]]; // _Marius A. Burtea_, Dec 03 2019

%o (PARI) a(n) = sum(k=1, n, binomial(n,k)*numdiv(k)*numdiv(n-k+1)); \\ _Michel Marcus_, Dec 05 2019

%Y Cf. A000005, A007318, A055507, A160399, A330088.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Dec 03 2019