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A306153
Inverse Weigh transform of (-1)^n * n!.
2
-1, 1, -4, 18, -92, 572, -4156, 34177, -314368, 3199844, -35703996, 433422067, -5687955724, 80256874912, -1211781887796, 19496946568897, -333041104402860, 6019770246910128, -114794574818830716, 2303332661419442477, -48509766592884311132, 1069983257387132347080
OFFSET
1,3
LINKS
N. J. A. Sloane, Transforms
FORMULA
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
EXAMPLE
(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 - ... .
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> (-1)^n*n!-b(n, n-1):
seq(a(n), n=1..24); # Alois P. Heinz, Jun 23 2018
MATHEMATICA
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}]]];
a[n_] := (-1)^n n! - b[n, n - 1] // FullSimplify;
Array[a, 24] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)
CROSSREFS
Cf. A168246.
Sequence in context: A337010 A081923 A020064 * A123589 A245103 A200717
KEYWORD
sign
AUTHOR
Seiichi Manyama, Jun 23 2018
STATUS
approved