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A293549
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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).
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10
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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
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OFFSET
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0,5
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COMMENTS
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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)
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LINKS
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FORMULA
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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).
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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