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).

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

Author

Ilya Gutkovskiy, Oct 11 2017

Keywords

• nonn

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

Crossrefs

Cf. A001222, A006171, A293548.

Programs

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}]

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).