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Inverse Weigh transform of n!.
6

%I #23 Nov 27 2020 03:50:41

%S 1,2,4,19,92,576,4156,34178,314368,3199936,35703996,433422071,

%T 5687955724,80256879068,1211781887796,19496946568898,333041104402860,

%U 6019770247224496,114794574818830716,2303332661419442569,48509766592884311132,1069983257387168051076

%N Inverse Weigh transform of n!.

%H Alois P. Heinz, <a href="/A168246/b168246.txt">Table of n, a(n) for n = 1..449</a>

%F Product_{k>=1} (1+x^k)^a(k) = Sum_{n>=0} n! x^n.

%F 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 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))

%p end:

%p a:= proc(n) option remember; n! -b(n, n-1) end:

%p seq(a(n), n=1..30); # _Alois P. Heinz_, Jun 11 2018

%t 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 a[n_] := a[n] = n! - b[n, n - 1];

%t Array[a, 30] (* _Jean-François Alcover_, Sep 16 2019, after _Alois P. Heinz_ *)

%o (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 Cf. A000142, A112354, A261052 (Weigh transform of n!).

%K easy,nonn

%O 1,2

%A _Vladeta Jovovic_, Nov 21 2009