A291552 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^k)^p(k), where p(k) is the number of partitions of k (A000041).

1, 2, 5, 11, 25, 52, 110, 221, 444, 868, 1685, 3212, 6082, 11361, 21071, 38693, 70570, 127670, 229557, 409963, 728069, 1285522, 2258318, 3947115, 6867238, 11893648, 20513199, 35235429, 60292928, 102787903, 174620017, 295644893, 498931699, 839367287, 1407864040, 2354559426, 3926878130

Offset: 0

Author

Ilya Gutkovskiy, Aug 30 2017

Keywords

• nonn

Comments

Partial sums of A001970.

Examples

			Equivalently (Cayley), a(n) = total number of 2-dimensional partitions of all nonnegative integers <= n.
a(3) = 11 because we have:
0...1...2.11.1...3.21.2.111.11.1
.............1........1.....1..1
...............................1
and 1 + 1 + 3 + 6 = 11.
		

Links

• Index entries for related partition-counting sequences

Crossrefs

Cf. A000041, A000070, A001970, A026905, A036469, A091360, A277643.

Programs

Maple

with(numtheory): with(combinat):
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      numbpart(d), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; b(n)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 11 2017

Mathematica

nmax = 36; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^k)^PartitionsP[k], {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: (1/(1 - x))*Product_{k>=1} 1/(1 - x^k)^p(k), where p(k) = [x^k] Product_{k>=1} 1/(1 - x^k).