A293549 - OEIS

%I #6 Sep 10 2021 06:20:10

%S 1,0,1,1,3,2,6,5,13,12,23,24,47,47,82,92,152,167,265,301,462,532,779,

%T 914,1324,1548,2174,2590,3573,4250,5771,6904,9254,11092,14638,17606,

%U 23043,27680,35820,43155,55383,66642,84850,102141,129171,155394,195134,234679,293184,352096,437359

%N 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).

%C Euler transform of A001222.

%C Comment from _R. J. Mathar_, Sep 10 2021 (Start):

%C The triangle of the multiset transformation of A001222 looks as follows:

%C 1 ;1

%C 0 0 ;0

%C 0 1 0 ;1

%C 0 1 0 0 ;1

%C 0 2 1 0 0 ;3

%C 0 1 1 0 0 0 ;2

%C 0 2 3 1 0 0 0 ;6

%C 0 1 3 1 0 0 0 0 ;5

%C 0 3 6 3 1 0 0 0 0 ;13

%C 0 2 5 4 1 0 0 0 0 0 ;12

%C 0 2 9 8 3 1 0 0 0 0 0 ;23

%C 0 1 9 9 4 1 0 0 0 0 0 0 ;24

%C 0 3 14 17 9 3 1 0 0 0 0 0 0 ;47

%C 0 1 12 18 11 4 1 0 0 0 0 0 0 0 ;47

%C 0 2 17 29 21 9 3 1 0 0 0 0 0 0 0 ;82

%C ...

%C The second column is A001222, the row sums (after the semicolons) are this sequence. (End)

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F 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)).

%F a(0) = 1; a(n) = (1/n)*Sum_{k=1..n} a(n-k)*b(k), b(k) = Sum_{d|k} d*bigomega(d).

%t nmax = 50; CoefficientList[Series[Product[1/(1 - x^k)^PrimeOmega[k], {k, 2, nmax}], {x, 0, nmax}], x]

%t 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}]

%Y Cf. A001222, A006171, A293548.

%K nonn

%O 0,5

%A _Ilya Gutkovskiy_, Oct 11 2017