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A302549 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} 1/(1 - x^(k*j))^j).

%I A302549 #12 Feb 16 2025 08:33:53
%S A302549 1,4,7,17,25,58,87,177,289,528,860,1550,2486,4257,6910,11474,18335,
%T A302549 29941,47331,75819,118887,187338,290784,452904,696058,1071234,1632947,
%U A302549 2487504,3759613,5676424,8512310,12744903,18975839,28194293,41691157,61516394,90379785
%N A302549 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} 1/(1 - x^(k*j))^j).
%C A302549 Inverse Moebius transform of A000219.
%H A302549 Alois P. Heinz, <a href="/A302549/b302549.txt">Table of n, a(n) for n = 1..10000</a>
%H A302549 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A302549 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PlanePartition.html">Plane Partition</a>
%F A302549 G.f.: Sum_{k>=1} A000219(k)*x^k/(1 - x^k).
%F A302549 a(n) = Sum_{d|n} A000219(d).
%p A302549 b:= proc(n) option remember; `if`(n=0, 1, add(
%p A302549       b(n-j)*numtheory[sigma][2](j), j=1..n)/n)
%p A302549     end:
%p A302549 a:= n-> add(b(d), d=numtheory[divisors](n)):
%p A302549 seq(a(n), n=1..40);  # _Alois P. Heinz_, Jun 21 2018
%t A302549 nmax = 37; Rest[CoefficientList[Series[Sum[-1 + Product[1/(1 - x^(k j))^j, {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]]
%t A302549 b[n_] := b[n] = SeriesCoefficient[Product[1/(1 - x^k)^k , {k, 1, n}], {x, 0, n}]; a[n_] := a[n] = SeriesCoefficient[Sum[b[k] x^k/(1 - x^k), {k, 1, n}], {x, 0, n}]; Table[a[n], {n, 37}]
%t A302549 b[0] = 1; b[n_] := b[n] = Sum[b[n - j] DivisorSigma[2, j], {j, n}]/n; a[n_] := a[n] = Sum[b[d], {d, Divisors[n]}]; Table[a[n], {n, 37}]
%Y A302549 Cf. A000219, A047966, A047968, A300275, A302550.
%K A302549 nonn
%O A302549 1,2
%A A302549 _Ilya Gutkovskiy_, Jun 20 2018