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A063835 Three times partitioned numbers: the number of ways a number can be partitioned in (not necessarily different) parts and each part again so partitioned a second and a third time.

%I A063835 #15 Mar 28 2016 04:06:12
%S A063835 1,5,14,51,125,429,1039,3258,8254,23554,58934,168803,412177,1114550,
%T A063835 2795446,7345875,18035424,46875324,114272057,291692396,709742614,
%U A063835 1774402071,4290848175,10672950659,25572179792,62670553073,149978278320
%N A063835 Three times partitioned numbers: the number of ways a number can be partitioned in (not necessarily different) parts and each part again so partitioned a second and a third time.
%H A063835 Vaclav Kotesovec, <a href="/A063835/b063835.txt">Table of n, a(n) for n = 1..2000</a>
%F A063835 G.f.: 1/Product(1-b(n)*x^n, n=1..infinity), where b(n) is sum of number of partitions of parts in all partitions of n; b() is convolution of A047968() and A000041(). - _Vladeta Jovovic_, Nov 22 2005
%F A063835 From _Vaclav Kotesovec_, Mar 28 2016: (Start)
%F A063835 a(n) ~ c * 21^(n/4), where
%F A063835 c = 31506.382471540934704971753670563958673161001663... if mod(n,4) = 0
%F A063835 c = 31502.248225846169487427060315658509213347537914... if mod(n,4) = 1
%F A063835 c = 31506.175349116205868096360427802563935891182649... if mod(n,4) = 2
%F A063835 c = 31502.232274793501377850265964413938565498517297... if mod(n,4) = 3
%F A063835 (End)
%t A063835 Table[Plus@@((Apply[Plus, #/. i_Integer-> PartitionsP[i], {1}]/. f->Times)& /@ Flatten[Flatten[Outer[f, Sequence@@(Partitions/@#), 1]]&/@Partitions[w]]), {w, 16}]
%t A063835 nmax = 40; A047968 = Table[Sum[PartitionsP[d], {d, Divisors[n]}], {n, 1, nmax}]; conv = Table[Sum[A047968[[j]]*PartitionsP[m - j], {j, 1, m}], {m, 1, nmax}]; A063835 = Rest[CoefficientList[Series[Product[1/(1 - conv[[k]]*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Mar 27 2016 *)
%Y A063835 Cf. A063834.
%K A063835 nonn,nice
%O A063835 1,2
%A A063835 _Wouter Meeussen_, Aug 21 2001
%E A063835 More terms from _Vladeta Jovovic_, Nov 22 2005