login
A305850
Weigh transform of the Bell numbers (A000110).
4
1, 1, 2, 7, 21, 78, 305, 1304, 6007, 29854, 159012, 904986, 5479078, 35150263, 238033523, 1695554145, 12663533586, 98881246850, 805128085616, 6820302066048, 59983405937707, 546690232627480, 5154757226832625, 50208266917662433, 504482106565647708
OFFSET
0,3
LINKS
FORMULA
G.f.: Product_{k>=1} (1+x^k)^Bell(k).
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(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[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[g[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 12 2022, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 11 2018
STATUS
approved