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A328301 Expansion of Product_{k>0} 1/(1 - x^(k^k)).

%I A328301 #50 Jul 31 2021 08:50:36
%S A328301 1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,8,9,9,9,10,11,
%T A328301 11,11,12,13,13,13,14,15,15,15,16,17,17,17,18,19,19,19,20,21,21,22,23,
%U A328301 24,24,25,26,27,27,28,29,30,30,31,32,33,33,34,35,36,36,37,38,39,39
%N A328301 Expansion of Product_{k>0} 1/(1 - x^(k^k)).
%C A328301 Also number of partitions of n into parts k^k for k > 0.
%H A328301 Alois P. Heinz, <a href="/A328301/b328301.txt">Table of n, a(n) for n = 0..10000</a>
%F A328301 G.f.: 1 + Sum_{n>0} x^(n^n) / Product_{k=1..n} (1 - x^(k^k)).
%e A328301 G.f.: 1 + x/(1-x) + x^4/((1-x)*(1-x^4)) + x^27/((1-x)*(1-x^4)*(1-x^27)) + ... .
%p A328301 b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A328301       b(n, i-1)+(p-> `if`(p>n, 0, b(n-p, i)))(i^i))
%p A328301     end:
%p A328301 a:= n-> `if`(n<2, 1, b(n, floor((t-> t/LambertW(t))(log(n))))):
%p A328301 seq(a(n), n=0..100);  # _Alois P. Heinz_, Oct 12 2019
%t A328301 b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + With[{p = i^i}, If[p > n, 0, b[n - p, i]]]];
%t A328301 a[n_] := If[n < 2, 1, b[n, Floor[PowerExpand[Log[n]/ProductLog[Log[n]]]]]];
%t A328301 a /@ Range[0, 100] (* _Jean-François Alcover_, Mar 12 2021, after _Alois P. Heinz_ *)
%o A328301 (PARI) my(N=99, x='x+O('x^N)); m=1; while(N>=m^m, m++); Vec(1/prod(k=1, m-1, 1-x^k^k))
%Y A328301 Cf. A000312, A001156, A003108, A046042, A064986, A328325.
%K A328301 nonn
%O A328301 0,5
%A A328301 _Seiichi Manyama_, Oct 12 2019