A353992
a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} d^(k/d + 1) )/k.
Original entry on oeis.org
1, 7, 41, 314, 2194, 22764, 195348, 2374224, 27940176, 384636960, 4673720160, 95522440320, 1323221996160, 23481816503040, 489968947641600, 10853692580505600, 190580382936115200, 5408424680491929600, 105077728210820198400, 3399507785578641408000
Offset: 1
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a[n_] := n! * Sum[DivisorSum[k, #^(k/# + 1) &]/k, {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Aug 06 2022 *)
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a(n) = n!*sum(k=1, n, sumdiv(k, d, d^(k/d+1))/k);
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a(n) = n!*sum(k=1, n, sumdiv(k, d, (k/d)^d/d));
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-k*x^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;
A356598
Expansion of e.g.f. ( Product_{k>0} 1/(1 - k * x^k) )^exp(x).
Original entry on oeis.org
1, 1, 8, 60, 606, 6795, 96145, 1458051, 25584020, 487911129, 10231475323, 230541036627, 5647620829862, 146760059424017, 4075332758190265, 119876230004510557, 3727336891407329320, 121841674696261466385, 4187995620589733257695, 150589951713517027739551
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)^exp(x)))
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a354340(n) = n!*sum(k=1, n, sumdiv(k, d, d^(k/d+1))/(k*(n-k)!));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354340(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A356578
Expansion of e.g.f. ( Product_{k>0} 1/(1 - k * x^k) )^x.
Original entry on oeis.org
1, 0, 2, 15, 92, 1050, 8514, 147000, 1546544, 29673000, 478186920, 9011752200, 178483287432, 4205087686800, 91775320005264, 2290742704668600, 63289842765692160, 1696665419122968000, 50287699532618564544, 1549916411848463721600
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)^x))
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a354848(n) = (n-1)!*sumdiv(n, d, d^(n/d+1));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*a354848(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;
Showing 1-4 of 4 results.