A356408
Expansion of e.g.f. ( Product_{k>0} 1/(1 - x^k/k) )^(1/(1-x)).
Original entry on oeis.org
1, 1, 5, 29, 216, 1919, 20012, 236977, 3145832, 46122546, 739703182, 12865212172, 241040899668, 4836265824740, 103410589256452, 2346358252787094, 56285005757022752, 1422783492250963296, 37790069818311971640, 1051924374853915254048
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-x^k/k)^(1/(1-x))))
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a356406(n) = n!*sum(k=1, n, sumdiv(k, d, 1/(d*(k/d)^d)));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356406(j)*binomial(i-1, j-1)*v[i-j+1])); v;
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)))
A356407
a(n) = n! * Sum_{k=1..n} Sum_{d|k} 1/(d * ((k/d)!)^d).
Original entry on oeis.org
1, 4, 15, 70, 375, 2411, 17598, 146490, 1359291, 13978597, 157393368, 1929989029, 25568858978, 364288345409, 5551537358188, 90142504077194, 1553345359200299, 28317316174307405, 544431381017568696, 11010510372888267555, 233653645911730002976
Offset: 1
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a(n) = n!*sum(k=1, n, sumdiv(k, d, 1/(d*(k/d)!^d)));
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-x^k/k!))/(1-x)))
A354339
a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} 1/(d * (k/d)^d) )/(n-k)!.
Original entry on oeis.org
1, 4, 13, 47, 188, 939, 5332, 36196, 279085, 2464592, 23591753, 259110191, 3030440580, 38874240339, 535736880460, 8027897509136, 126034992483809, 2144006461602308, 38072688073456557, 723023026186433271, 14342481336066795732, 301141522554921194275
Offset: 1
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a308345(n) = n!*sumdiv(n, d, 1/(d*(n/d)^d));
a(n) = sum(k=1, n, a308345(k)*binomial(n, k));
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-exp(x)*sum(k=1, N, log(1-x^k/k))))
Showing 1-4 of 4 results.