OFFSET
0,5
COMMENTS
Euler transform of A001222.
Comment from R. J. Mathar, Sep 10 2021 (Start):
The triangle of the multiset transformation of A001222 looks as follows:
1 ;1
0 0 ;0
0 1 0 ;1
0 1 0 0 ;1
0 2 1 0 0 ;3
0 1 1 0 0 0 ;2
0 2 3 1 0 0 0 ;6
0 1 3 1 0 0 0 0 ;5
0 3 6 3 1 0 0 0 0 ;13
0 2 5 4 1 0 0 0 0 0 ;12
0 2 9 8 3 1 0 0 0 0 0 ;23
0 1 9 9 4 1 0 0 0 0 0 0 ;24
0 3 14 17 9 3 1 0 0 0 0 0 0 ;47
0 1 12 18 11 4 1 0 0 0 0 0 0 0 ;47
0 2 17 29 21 9 3 1 0 0 0 0 0 0 0 ;82
...
The second column is A001222, the row sums (after the semicolons) are this sequence. (End)
LINKS
N. J. A. Sloane, Transforms
FORMULA
G.f.: Product_{k>=2} 1/(1 - x^k)^b(k), where b(k) = [x^k] Sum_{p prime, j>=1} x^(p^j)/(1 - x^(p^j)).
a(0) = 1; a(n) = (1/n)*Sum_{k=1..n} a(n-k)*b(k), b(k) = Sum_{d|k} d*bigomega(d).
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[1/(1 - x^k)^PrimeOmega[k], {k, 2, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d PrimeOmega[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 50}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Oct 11 2017
STATUS
approved