%I #9 Apr 03 2019 02:59:03

%S 1,3,6,10,16,33,44,74,126,204,289,503,696,1151,1749,2599,3742,5928,

%T 8245,12658,18351,26715,37828,55296,78346,111882,159664,226782,315416,

%U 446670,618667,860764,1199995,1649820,2289020,3157349,4303996,5878786,8033272,10894516,14749052

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

%C Inverse Moebius transform of A022629.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F G.f.: Sum_{k>=1} A022629(k)*x^k/(1 - x^k).

%F a(n) = Sum_{d|n} A022629(d).

%p a:=series(add(-1+mul(1+j*x^(k*j),j=1..100),k=1..100),x=0,42): seq(coeff(a,x,n),n=1..41); # _Paolo P. Lava_, Apr 02 2019

%t nmax = 41; Rest[CoefficientList[Series[Sum[-1 + Product[(1 + j x^(k j)), {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]]

%t b[n_] := b[n] = SeriesCoefficient[Product[(1 + k x^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, 41}]

%t b[0] = 1; b[n_] := b[n] = Sum[Sum[(-d)^(k/d + 1), {d, Divisors[k]}] b[n - k], {k, 1, n}]/n; a[n_] := a[n] = Sum[b[d], {d, Divisors[n]}]; Table[a[n], {n, 41}]

%Y Cf. A022629, A047966, A300278, A302550, A318025.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Aug 23 2018