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A291552 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^k)^p(k), where p(k) is the number of partitions of k (A000041).

%I A291552 #16 Sep 11 2017 08:55:45
%S A291552 1,2,5,11,25,52,110,221,444,868,1685,3212,6082,11361,21071,38693,
%T A291552 70570,127670,229557,409963,728069,1285522,2258318,3947115,6867238,
%U A291552 11893648,20513199,35235429,60292928,102787903,174620017,295644893,498931699,839367287,1407864040,2354559426,3926878130
%N A291552 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^k)^p(k), where p(k) is the number of partitions of k (A000041).
%C A291552 Partial sums of A001970.
%H A291552 <a href="/index/Par#part">Index entries for related partition-counting sequences</a>
%F A291552 G.f.: (1/(1 - x))*Product_{k>=1} 1/(1 - x^k)^p(k), where p(k) = [x^k] Product_{k>=1} 1/(1 - x^k).
%e A291552 Equivalently (Cayley), a(n) = total number of 2-dimensional partitions of all nonnegative integers <= n.
%e A291552 a(3) = 11 because we have:
%e A291552 0...1...2.11.1...3.21.2.111.11.1
%e A291552 .............1........1.....1..1
%e A291552 ...............................1
%e A291552 and 1 + 1 + 3 + 6 = 11.
%p A291552 with(numtheory): with(combinat):
%p A291552 b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A291552       numbpart(d), d=divisors(j))*b(n-j), j=1..n)/n)
%p A291552     end:
%p A291552 a:= proc(n) option remember; b(n)+`if`(n>0, a(n-1), 0) end:
%p A291552 seq(a(n), n=0..40);  # _Alois P. Heinz_, Sep 11 2017
%t A291552 nmax = 36; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^k)^PartitionsP[k], {k, 1, nmax}], {x, 0, nmax}], x]
%Y A291552 Cf. A000041, A000070, A001970, A026905, A036469, A091360, A277643.
%K A291552 nonn
%O A291552 0,2
%A A291552 _Ilya Gutkovskiy_, Aug 30 2017