A305846 Inverse Weigh transform of the Bell numbers (A000110).

1, 2, 3, 11, 34, 138, 610, 2976, 15612, 87905, 526274, 3334988, 22270254, 156173299, 1146640394, 8791427525, 70227355786, 583283756678, 5027823752930, 44903579714037, 414877600876638, 3959945233249877, 38996757506464858, 395749369601741015, 4134132167178705732

Offset

1,2

Links

Alois P. Heinz, Table of n, a(n) for n = 1..576

Formula

Product_{k>=1} (1+x^k)^a(k) = Sum_{n>=0} Bell(n) * x^n.

Maple

g:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*g(n-j), j=1..n))
    end:
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:= proc(n) option remember; g(n)-b(n, n-1) end:
seq(a(n), n=1..30);

Mathematica

g[n_] := g[n] = If[n == 0, 1,
     Sum[Binomial[n-1, j-1]*g[n-j], {j, 1, n}]];
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_] := a[n] = g[n] - b[n, n - 1];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 10 2022, after Alois P. Heinz *)

Crossrefs

Cf. A000110, A085686, A168246, A305850, A305853.

Sequence in context: A300771 A296005 A159458 * A057838 A219497 A348126

Adjacent sequences: A305843 A305844 A305845 * A305847 A305848 A305849

Keyword

nonn

Author

Alois P. Heinz, Jun 11 2018

Status

approved