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A306009 Inverse Weigh transform of A000085.

%I A306009 #12 Oct 27 2021 09:27:27
%S A306009 1,2,2,7,14,43,130,446,1544,5773,22170,89356,370198,1591379,7020014,
%T A306009 31922981,148679262,710828036,3474337098,17379964444,88739068866,
%U A306009 462670294023,2458638559154,13317850411827,73432568553848,412120738922369,2351720323257872
%N A306009 Inverse Weigh transform of A000085.
%H A306009 Alois P. Heinz, <a href="/A306009/b306009.txt">Table of n, a(n) for n = 1..800</a>
%F A306009 Product_{k>=1} (1+x^k)^a(k) = Sum_{n>=0} A000085(n) * x^n.
%p A306009 g:= proc(n) option remember;
%p A306009       `if`(n<2, 1, g(n-1)+(n-1)*g(n-2))
%p A306009     end:
%p A306009 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A306009       add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
%p A306009     end:
%p A306009 a:= proc(n) option remember; g(n)-b(n, n-1) end:
%p A306009 seq(a(n), n=1..30);
%t A306009 g[n_] := g[n] = If[n < 2, 1, g[n - 1] + (n - 1)*g[n - 2]];
%t A306009 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
%t A306009 a[n_] := a[n] = g[n] - b[n, n - 1];
%t A306009 Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Oct 27 2021, after _Alois P. Heinz_ *)
%Y A306009 Cf. A000085, A293114.
%K A306009 nonn
%O A306009 1,2
%A A306009 _Alois P. Heinz_, Jun 16 2018