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

1, 4, 7, 17, 25, 58, 87, 177, 289, 528, 860, 1550, 2486, 4257, 6910, 11474, 18335, 29941, 47331, 75819, 118887, 187338, 290784, 452904, 696058, 1071234, 1632947, 2487504, 3759613, 5676424, 8512310, 12744903, 18975839, 28194293, 41691157, 61516394, 90379785

Offset: 1

Author

Ilya Gutkovskiy, Jun 20 2018

Keywords

• nonn

Comments

Inverse Moebius transform of A000219.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• N. J. A. Sloane, Transforms
• Eric Weisstein's World of Mathematics, Plane Partition

Crossrefs

Cf. A000219, A047966, A047968, A300275, A302550.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> add(b(d), d=numtheory[divisors](n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Jun 21 2018

Mathematica

nmax = 37; Rest[CoefficientList[Series[Sum[-1 + Product[1/(1 - x^(k j))^j, {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]]
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}]
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}]

Formula

G.f.: Sum_{k>=1} A000219(k)*x^k/(1 - x^k).

a(n) = Sum_{d|n} A000219(d).