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%I #4 May 26 2018 08:46:04
%S 1,1,3,14,79,539,4663,42468,457945,5433281,71036231,994289658,
%T 15544425103,253283689619,4489180389835,84521336758904,
%U 1687130833152561,35365641206048129,790065486354237643,18340253632236738022,449655289227002010351,11492300073384698090795,306803167368168113022271
%N Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^AH(k), where AH(k) is the k-th alternating harmonic number.
%C a(n)/n! is the Euler transform of [1, 1 - 1/2, 1 - 1/2 + 1/3, 1 - 1/2 + 1/3 - 1/4, 1 - 1/2 + 1/3 - 1/4 + 1/5, ...].
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F E.g.f.: Product_{k>=1} 1/(1 - x^k)^(A058313(k)/A058312(k)).
%t nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^Sum[(-1)^(j + 1)/j, {j, 1, k}], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d ((-1)^(d + 1) LerchPhi[-1, 1, d + 1] + Log[2]), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]
%Y Cf. A028342, A058312, A058313, A168243, A303970.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, May 26 2018
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