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A305852
Weigh transform of the Fubini numbers (ordered Bell numbers, A000670).
3
1, 1, 3, 16, 91, 658, 5567, 54917, 620081, 7905592, 112382245, 1762646331, 30231516786, 562750751610, 11297034281595, 243241826522376, 5591075279423398, 136633359995403580, 3537193288612096901, 96697587673174195740, 2783492094736121087958
OFFSET
0,3
LINKS
FORMULA
G.f.: Product_{k>=1} (1+x^k)^A000670(k).
a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Sep 10 2019
MAPLE
g:= proc(n) option remember; `if`(n=0, 1,
add(g(n-j)*binomial(n, j), j=1..n))
end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(g(i), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30);
MATHEMATICA
g[n_] := g[n] = If[n == 0, 1,
Sum[g[n - j] Binomial[n, j], {j, 1, n}]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0,
Sum[Binomial[g[i], j] b[n - i j, i - 1], {j, 0, n/i}]]];
a[n_] := b[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 11 2018
STATUS
approved