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A308272 G.f. A(x) satisfies: A(x) = (1 + x) * A(x^2)*A(x^3)*A(x^5)* ... *A(x^prime(k))* ...

%I A308272 #7 May 17 2019 22:08:11
%S A308272 1,1,1,2,2,3,5,6,7,10,13,16,22,27,33,44,53,65,84,101,124,156,187,226,
%T A308272 280,336,403,492,587,700,850,1008,1195,1435,1693,2004,2390,2808,3303,
%U A308272 3910,4584,5372,6328,7387,8619,10106,11757,13675,15961,18508,21464,24948,28845,33345
%N A308272 G.f. A(x) satisfies: A(x) = (1 + x) * A(x^2)*A(x^3)*A(x^5)* ... *A(x^prime(k))* ...
%C A308272 Weigh transform of A008480.
%F A308272 G.f.: Product_{k>=1} (1 + x^k)^A008480(k).
%p A308272 g:= proc(n) option remember; (l-> add(i, i=l)!/
%p A308272       mul(i!, i=l))(map(i-> i[2], ifactors(n)[2]))
%p A308272     end:
%p A308272 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A308272       add(binomial(g(i), j)*b(n-i*j, i-1), j=0..n/i)))
%p A308272     end:
%p A308272 a:= n-> b(n$2):
%p A308272 seq(a(n), n=0..60);  # _Alois P. Heinz_, May 17 2019
%t A308272 terms = 53; A[_] = 1; Do[A[x_] = (1 + x) Product[A[x^Prime[k]], {k, 1, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
%Y A308272 Cf. A000040, A008480, A129373, A308271.
%K A308272 nonn
%O A308272 0,4
%A A308272 _Ilya Gutkovskiy_, May 17 2019