A296116 Number of partitions in which each summand, s, may be used with frequency f if f divides s.

1, 1, 1, 2, 3, 4, 4, 6, 9, 12, 14, 18, 23, 29, 35, 43, 56, 68, 82, 100, 122, 147, 174, 209, 252, 302, 356, 421, 500, 589, 690, 808, 952, 1110, 1292, 1505, 1756, 2034, 2348, 2715, 3139, 3620, 4156, 4778, 5492, 6296, 7195, 8220, 9398, 10714, 12194, 13872, 15784

Offset: 0

Author

David S. Newman, Dec 04 2017

Keywords

• nonn

Examples

			For n=3, the partitions counted are 3 and 2+1.
For n=4: 4, 3+1, 2+2.
For n=5: 5, 4+1, 3+2, 2+2+1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Index entries for related partition-counting sequences

Crossrefs

Cf. A100471, A100881, A100882, A100883.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1 or n<0, 0,
      b(n, i-1)+add(b(n-i*j, i-1), j=numtheory[divisors](i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 05 2017

Mathematica

iend = 30;
s = Series[Product[1 + Sum[x^(Divisors[n][[i]] n), {i, 1, Length[Divisors[n]]}], {n, 1, iend}], {x, 0, iend}]; Print[s];
CoefficientList[s, x]

Formula

G.f.: Product_{n >= 1} (1 + Sum_{d divides n} x^(d*n)).

Extensions

More terms from Alois P. Heinz, Dec 05 2017