A306152 Inverse Weigh transform of n^n.

1, 4, 23, 227, 2800, 42599, 763220, 15734615, 366715248, 9533820200, 273549419552, 8586984284469, 292755986184548, 10772849584162694, 425587711650564816, 17966217347001535765, 807152054953801845760, 38451365602113718874568, 1936082850634342992601636

Offset: 1

Author

Seiichi Manyama, Jun 23 2018

Keywords

• nonn

Examples

			(1+x)*(1+x^2)^4*(1+x^3)^23*(1+x^4)^227* ... = 1 + x + 4*x^2 + 27*x^3 + 256*x^4 + ... .
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..386
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000312, A306154.

Programs

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-> 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_] := n^n - b[n, n - 1];
Table[a[n], {n, 1, 24}] (* Jean-François Alcover, Mar 19 2022, after Alois P. Heinz *)

Formula

a(n) ~ n^n. - Vaclav Kotesovec, Mar 19 2022