A304967 Expansion of Product_{k>=1} 1/(1 - x^k)^(p(k)-p(k-1)), where p(k) = number of partitions of k (A000041).

1, 0, 1, 1, 3, 3, 8, 9, 20, 26, 49, 68, 123, 173, 295, 432, 707, 1044, 1672, 2483, 3900, 5817, 8993, 13424, 20539, 30609, 46399, 69052, 103879, 154198, 230550, 341261, 507484, 749028, 1108559, 1631340, 2404311, 3527615, 5179317, 7577263, 11086413, 16173577, 23588227

Offset

0,5

Comments

Euler transform of A002865.

Links

Table of n, a(n) for n=0..42.

N. J. A. Sloane, Transforms

Index entries for sequences related to partitions

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A002865(k).

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(
      (numtheory[sigma](j)-1)*b(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      b(d), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, May 22 2018

Mathematica

nmax = 42; CoefficientList[Series[Product[1/(1 - x^k)^(PartitionsP[k] - PartitionsP[k - 1]), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (PartitionsP[d] - PartitionsP[d - 1]), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 42}]

Crossrefs

Cf. A000041, A000219, A001383, A001970, A002865, A304966.

Sequence in context: A092549 A260890 A022663 * A323654 A092481 A099508

Adjacent sequences: A304964 A304965 A304966 * A304968 A304969 A304970

Keyword

nonn

Author

Ilya Gutkovskiy, May 22 2018

Status

approved