A308337 -id:A308337 - cp's OEIS frontend

A308345 Expansion of e.g.f. Sum_{k>=1} log(1/(1 - x^k/k)).

1, 2, 4, 15, 48, 310, 1440, 11970, 85120, 821016, 7257600, 91707000, 958003200, 13440913200, 178919989248, 2809456650000, 41845579776000, 763629026160000, 12804747411456000, 257140635922025856, 4918792391884800000, 106876408948152480000

Offset: 1

Ilya Gutkovskiy, May 21 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..450

Cf. A007841, A038048, A182926, A182927, A308337.

nmax = 22; CoefficientList[Series[Sum[Log[1/(1 - x^k/k)], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[n! Sum[1/(d (n/d)^d), {d, Divisors[n]}], {n, 1, 22}]

a(n) = n! * Sum_{d|n} 1/(d*(n/d)^d).
a(n) = A007841(n) - (1/n) * Sum_{k=1..n-1} k*binomial(n,k)*A007841(n-k)*a(k).
a(n) ~ 2 * (n-1)!. - Vaclav Kotesovec, Feb 16 2020

A308338 Expansion of e.g.f. exp(-1 + Product_{k>=1} (1 + x^k/k)).

Original entry on oeis.org

1, 1, 2, 9, 44, 270, 2064, 17682, 171296, 1867968, 22470840, 294493320, 4195969392, 64416698112, 1059685905264, 18609306423120, 347179119075840, 6855335163907200, 142889687354283264, 3133647091691585280, 72124075333003155840, 1738384773846440146560

Offset: 0

Ilya Gutkovskiy, May 20 2019

nonn

Cf. A007838, A308336, A308337.

nmax = 21; CoefficientList[Series[Exp[Product[(1 + x^k/k), {k, 1, nmax}] - 1], {x, 0, nmax}], x] Range[0, nmax]!

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1)*A007838(k)*a(n-k).

cp's OEIS Frontend

A308345 Expansion of e.g.f. Sum_{k>=1} log(1/(1 - x^k/k)).

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