A306153 - OEIS

%I #15 Nov 27 2020 04:07:13

%S -1,1,-4,18,-92,572,-4156,34177,-314368,3199844,-35703996,433422067,

%T -5687955724,80256874912,-1211781887796,19496946568897,

%U -333041104402860,6019770246910128,-114794574818830716,2303332661419442477,-48509766592884311132,1069983257387132347080

%N Inverse Weigh transform of (-1)^n * n!.

%H Alois P. Heinz, <a href="/A306153/b306153.txt">Table of n, a(n) for n = 1..449</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F a(n) ~ (-1)^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

%e (1+x)^(-1)*(1+x^2)*(1+x^3)^(-4)*(1+x^4)^18* ... = 1 - x + 2*x^2 - 6*x^3 + 24*x^4 - ... .

%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:= n-> (-1)^n*n!-b(n, n-1):

%p seq(a(n), n=1..24);  # _Alois P. Heinz_, Jun 23 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_] := (-1)^n n! - b[n, n - 1] // FullSimplify;

%t Array[a, 24] (* _Jean-François Alcover_, Nov 27 2020, after _Alois P. Heinz_ *)

%Y Cf. A168246.

%K sign

%O 1,3

%A _Seiichi Manyama_, Jun 23 2018