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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 (list; graph; refs; listen; history; text; internal format)
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
Sequence in context: A301501 A072787 A338524 * A370377 A306443 A336518
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Oct 11 2017
STATUS
approved

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Last modified February 21 04:30 EST 2024. Contains 370219 sequences. (Running on oeis4.)