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A293549
Expansion of Product_{k>=2} 1/(1 - x^k)^bigomega(k), where bigomega(k) is the number of prime divisors of k counted with multiplicity (A001222).
10
1, 0, 1, 1, 3, 2, 6, 5, 13, 12, 23, 24, 47, 47, 82, 92, 152, 167, 265, 301, 462, 532, 779, 914, 1324, 1548, 2174, 2590, 3573, 4250, 5771, 6904, 9254, 11092, 14638, 17606, 23043, 27680, 35820, 43155, 55383, 66642, 84850, 102141, 129171, 155394, 195134, 234679, 293184, 352096, 437359
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
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