A304967 - OEIS

%I #7 May 22 2018 20:34:10

%S 1,0,1,1,3,3,8,9,20,26,49,68,123,173,295,432,707,1044,1672,2483,3900,

%T 5817,8993,13424,20539,30609,46399,69052,103879,154198,230550,341261,

%U 507484,749028,1108559,1631340,2404311,3527615,5179317,7577263,11086413,16173577,23588227

%N Expansion of Product_{k>=1} 1/(1 - x^k)^(p(k)-p(k-1)), where p(k) = number of partitions of k (A000041).

%C Euler transform of A002865.

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

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F G.f.: Product_{k>=1} 1/(1 - x^k)^A002865(k).

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

%p       (numtheory[sigma](j)-1)*b(n-j), j=1..n)/n)

%p     end:

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

%p       b(d), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..50);  # _Alois P. Heinz_, May 22 2018

%t nmax = 42; CoefficientList[Series[Product[1/(1 - x^k)^(PartitionsP[k] - PartitionsP[k - 1]), {k, 1, nmax}], {x, 0, nmax}], x]

%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (PartitionsP[d] - PartitionsP[d - 1]), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 42}]

%Y Cf. A000041, A000219, A001383, A001970, A002865, A304966.

%K nonn

%O 0,5

%A _Ilya Gutkovskiy_, May 22 2018