A305049 - OEIS

%I #5 May 24 2018 20:03:54

%S 1,1,3,8,27,67,216,569,1747,4812,14041,39483,115408,326385,941735,

%T 2684170,7725097,22063737,63354066,181223899,519883185,1488316952,

%U 4266788191,12219763777,35023995792,100326757107,287503501905,823654031283,2360146144917,6761847714698,19374935267810

%N Expansion of 1/(1 - Sum_{k>=1} tau_k(k)*x^k), where tau_k(k) = number of ordered k-factorizations of k (A163767).

%C Invert transform of A163767.

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

%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>

%F G.f.: 1/(1 - Sum_{k>=1} A163767(k)*x^k).

%p A:= proc(n, k) option remember; `if`(k=1, 1,

%p       add(A(d, k-1), d=numtheory[divisors](n)))

%p     end:

%p a:= proc(n) option remember; `if`(n=0, 1,

%p       add(A(j$2)*a(n-j), j=1..n))

%p     end:

%p seq(a(n), n=0..35);  # _Alois P. Heinz_, May 24 2018

%t nmax = 30; CoefficientList[Series[1/(1 - Sum[Times @@ (Binomial[# + k - 1, k - 1] & /@ FactorInteger[k][[All, 2]]) x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

%t a[0] = 1; a[n_] := a[n] = Sum[Times @@ (Binomial[# + k - 1, k - 1] & /@ FactorInteger[k][[All, 2]]) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 30}]

%Y Cf. A001906, A055887, A088305, A129373, A129374, A129921, A163767, A304963, A304964, A304965.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, May 24 2018