login
A296116
Number of partitions in which each summand, s, may be used with frequency f if f divides s.
1
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
0,4
FORMULA
G.f.: Product_{n >= 1} (1 + Sum_{d divides n} x^(d*n)).
EXAMPLE
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.
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
David S. Newman, Dec 04 2017
EXTENSIONS
More terms from Alois P. Heinz, Dec 05 2017
STATUS
approved