A356437
a(n) = n! * Sum_{k=1..n} sigma_k(k)/k.
Original entry on oeis.org
1, 7, 77, 1946, 84754, 6202524, 636369348, 89979720144, 16431405256656, 3796658174518560, 1077102230236529760, 368915006390671969920, 149873555740938949215360, 71297150722148582901815040, 39244301012876892023553235200
Offset: 1
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Table[n! * Sum[DivisorSigma[k, k]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 07 2022 *)
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a(n) = n!*sum(k=1, n, sigma(k, k)/k);
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my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-(k*x)^k)/k)/(1-x)))
A356439
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, 6, 39, 344, 3410, 42234, 567126, 8812880, 149409144, 2793232440, 56224856160, 1234342760232, 28773852409848, 718719835537872, 19045601930731320, 534564416062012800, 15792205306586537280, 491639547448322794944, 16024048206145815040704
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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a356436(n) = n!*sum(k=1, n, sumdiv(k, d, d^(k/d))/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356436(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A356587
Expansion of e.g.f. ( Product_{k>0} 1/(1 - (k * x)^k)^(1/k) )^x.
Original entry on oeis.org
1, 0, 2, 15, 236, 8490, 459234, 40325880, 4777773104, 767688946920, 156746202491880, 40056474754165320, 12448131138826294152, 4634982982962988690320, 2033625840922821008112144, 1039060311676326627685615800, 611331728108400284878223051520
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))^x))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*sigma(j-1, j-1)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;
Showing 1-3 of 3 results.