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 A305305 Expansion of e.g.f. 1/(1 - Sum_{k>=1} x^k/(k*(1 - x^k))). 1
 1, 1, 5, 32, 292, 3174, 42758, 659028, 11725656, 233646240, 5183599152, 126353158656, 3362529785712, 96896454983184, 3007687250735568, 100017757744279584, 3547903924884082176, 133715849506895518848, 5336112511923188151168, 224772952826373341478912, 9966476790792153522756864 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n)/n! is the invert transform of [1, 3/2, 4/3, 7/4, 6/5, ... = sums of reciprocals of divisors of 1, 2, 3, 4, 5, ...]. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..397 N. J. A. Sloane, Transforms FORMULA E.g.f.: 1/(1 - Sum_{k>=1} (sigma(k)/k)*x^k), where sigma() = A000203. E.g.f.: 1/(1 - Sum_{k>=1} (A017665(k)/A017666(k))*x^k). E.g.f.: 1/(1 - log(f(x))), where f(x) = o.g.f. for A000041, Product_{k>=1} 1/(1 - x^k). EXAMPLE E.g.f.: A(x) = 1 + x + 5*x^2/2! + 32*x^3/3! + 292*x^4/4! + 3174*x^5/5! + 42758*x^6/6! + ... MAPLE b:= proc(n) option remember; `if`(n=0, 1, add(add(       1/d, d=numtheory[divisors](j))*b(n-j), j=1..n))     end: a:= n-> b(n)*n!: seq(a(n), n=0..20);  # Alois P. Heinz, May 29 2018 MATHEMATICA nmax = 20; CoefficientList[Series[1/(1 - Sum[x^k/(k (1 - x^k)), {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]! nmax = 20; CoefficientList[Series[1/(1 - Sum[DivisorSigma[-1, k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]! a[0] = 1; a[n_] := a[n] = Sum[DivisorSigma[-1, k] a[n - k], {k, 1, n}]; Table[n! a[n], {n, 0, 20}] CROSSREFS Cf. A000041, A000203, A017665, A017666, A038048, A180305. Sequence in context: A331660 A001923 A257710 * A331339 A307497 A023880 Adjacent sequences:  A305302 A305303 A305304 * A305306 A305307 A305308 KEYWORD nonn AUTHOR Ilya Gutkovskiy, May 29 2018 STATUS approved

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Last modified January 26 23:05 EST 2020. Contains 331289 sequences. (Running on oeis4.)