A356436
a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} d^(k/d) )/k.
Original entry on oeis.org
1, 5, 23, 146, 874, 8124, 62628, 707664, 7860816, 103284000, 1179669600, 24454569600, 324615427200, 5740203974400, 119579523436800, 2688723275212800, 46084905896601600, 1383333631684300800, 26411386476116275200, 868104140064602112000
Offset: 1
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a(n) = n!*sum(k=1, n, sumdiv(k, d, d^(k/d))/k);
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-k*x^k)/k)/(1-x)))
A356440
Expansion of e.g.f. ( Product_{k>0} 1/(1 - (k * x)^k)^(1/k) )^(1/(1-x)).
Original entry on oeis.org
1, 1, 8, 99, 2444, 101450, 7045194, 701736966, 97147459184, 17505366041880, 4005462950166600, 1128394974054308400, 384386423684496873672, 155497732356686080354968, 73718160600338917089657216, 40462026280443230503858113240
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-(k*x)^k)^(1/k))^(1/(1-x))))
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a356437(n) = n!*sum(k=1, n, sigma(k, k)/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356437(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A356589
a(n) = n! * Sum_{k=1..n} sigma_k(k)/(k * (n-k)!).
Original entry on oeis.org
1, 7, 74, 1896, 83829, 6169915, 634444586, 89796130088, 16407420884385, 3792452363345383, 1076168167972120354, 368657061467873013440, 149787334364400115372677, 71262783791831946810277899, 39228224120114488162020163762
Offset: 1
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a[n_] := n! * Sum[DivisorSigma[k, k]/(k*(n - k)!), {k, 1, n}]; Array[a, 15] (* Amiram Eldar, Aug 14 2022 *)
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a(n) = n!*sum(k=1, n, sigma(k, k)/(k*(n-k)!));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(-exp(x)*sum(k=1, N, log(1-(k*x)^k)/k)))
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