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A168246 Inverse Weigh transform of n!.

%I A168246 #23 Nov 27 2020 03:50:41
%S A168246 1,2,4,19,92,576,4156,34178,314368,3199936,35703996,433422071,
%T A168246 5687955724,80256879068,1211781887796,19496946568898,333041104402860,
%U A168246 6019770247224496,114794574818830716,2303332661419442569,48509766592884311132,1069983257387168051076
%N A168246 Inverse Weigh transform of n!.
%H A168246 Alois P. Heinz, <a href="/A168246/b168246.txt">Table of n, a(n) for n = 1..449</a>
%F A168246 Product_{k>=1} (1+x^k)^a(k) = Sum_{n>=0} n! x^n.
%F A168246 a(n) ~ n! * (1 - 1/n - 1/n^2 - 4/n^3 - 23/n^4 - 171/n^5 - 1542/n^6 - 16241/n^7 - 194973/n^8 - 2622610/n^9 - 39027573/n^10 - ...), for coefficients see A113869. - _Vaclav Kotesovec_, Nov 27 2020
%p A168246 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A168246       add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
%p A168246     end:
%p A168246 a:= proc(n) option remember; n! -b(n, n-1) end:
%p A168246 seq(a(n), n=1..30);  # _Alois P. Heinz_, Jun 11 2018
%t A168246 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 A168246 a[n_] := a[n] = n! - b[n, n - 1];
%t A168246 Array[a, 30] (* _Jean-François Alcover_, Sep 16 2019, after _Alois P. Heinz_ *)
%o A168246 (PARI) seq(n)={dirdiv(Vec(log(1+x*Ser(vector(n, n, n!)))), -vector(n, n, (-1)^n/n))} \\ _Andrew Howroyd_, Jun 22 2018
%Y A168246 Cf. A000142, A112354, A261052 (Weigh transform of n!).
%K A168246 easy,nonn
%O A168246 1,2
%A A168246 _Vladeta Jovovic_, Nov 21 2009