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A328325 Expansion of Product_{k>=0} 1/(1 - x^(k^k)).

%I A328325 #35 Sep 14 2022 07:26:24
%S A328325 1,2,3,4,6,8,10,12,15,18,21,24,28,32,36,40,45,50,55,60,66,72,78,84,91,
%T A328325 98,105,113,122,131,140,150,161,172,183,195,208,221,234,248,263,278,
%U A328325 293,309,326,343,360,378,397,416,435,455,476,497,519,542,566,590,615,641,668,695
%N A328325 Expansion of Product_{k>=0} 1/(1 - x^(k^k)).
%C A328325 Partial sums of A328301.
%H A328325 Alois P. Heinz, <a href="/A328325/b328325.txt">Table of n, a(n) for n = 0..10000</a>
%F A328325 a(n) = Sum_{k=0..n} A328301(k).
%F A328325 G.f.: 1/(1-x) + Sum_{n>0} x^(n^n) / Product_{k=0..n} (1 - x^(k^k)).
%e A328325 G.f.: 1/(1-x) + x/(1-x)^2 + x^4/((1-x)^2*(1-x^4)) + x^27/((1-x)^2*(1-x^4)*(1-x^27)) + ... .
%p A328325 b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A328325       b(n, i-1)+(p-> `if`(p>n, 0, b(n-p, i)))(i^i))
%p A328325     end:
%p A328325 a:= proc(n) option remember; `if`(n<2, n+1, a(n-1)+
%p A328325       b(n, floor((t-> t/LambertW(t))(log(n)))))
%p A328325     end:
%p A328325 seq(a(n), n=0..100);  # _Alois P. Heinz_, Oct 12 2019
%t A328325 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 A328325 a[n_] := a[n] = If[n < 2, n+1, a[n-1] + b[n, Floor[PowerExpand[Log[n]/ ProductLog[Log[n]]]]]];
%t A328325 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Sep 14 2022, after _Alois P. Heinz_ *)
%o A328325 (PARI) N=99; x='x+O('x^N); m=1; while(N>=m^m, m++); Vec(1/prod(k=0, m-1, 1-x^k^k))
%Y A328325 Cf. A000312, A064985, A086858, A328301.
%K A328325 nonn
%O A328325 0,2
%A A328325 _Seiichi Manyama_, Oct 12 2019