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A317911
Expansion of Product_{k>=2} 1/(1 - x^k)^p(k), where p(k) = number of partitions of k (A000041).
1
1, 0, 2, 3, 8, 13, 31, 53, 112, 201, 393, 710, 1343, 2409, 4431, 7912, 14255, 25223, 44787, 78519, 137700, 239347, 415343, 716001, 1231326, 2106287, 3593141, 6102679, 10335269, 17437476, 29337139, 49192762, 82261930, 137148782, 228061165, 378198633, 625623318, 1032301633
OFFSET
0,3
COMMENTS
First differences of A001970.
FORMULA
G.f.: exp(Sum_{j>=1} Sum_{k>=2} p(k)*x^(j*k)/j).
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:= n-> b(n)-b(n-1):
seq(a(n), n=0..40); # Alois P. Heinz, Aug 10 2018
MATHEMATICA
nmax = 37; CoefficientList[Series[Product[1/(1 - x^k)^PartitionsP[k], {k, 2, nmax}], {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[Exp[Sum[Sum[PartitionsP[k] x^(j k)/j, {k, 2, nmax}], {j, 1, nmax}]], {x, 0, nmax}], x]
b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d PartitionsP[d], {d, Divisors[k]}] b[n - k], {k, 1, n}]/n]; Differences[Table[b[n], {n, -1, 37}]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Aug 10 2018
STATUS
approved