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Euler transform of the Bell numbers (A000110).
3

%I #11 Jun 11 2018 18:30:26

%S 1,1,3,8,26,88,340,1411,6417,31474,166242,939646,5659613,36158227,

%T 244049562,1733702757,12919475840,100690425442,818554392962,

%U 6924577964036,60828588178031,553821749290234,5217264062756556,50776256646839085,509823607380230570

%N Euler transform of the Bell numbers (A000110).

%H Alois P. Heinz, <a href="/A290351/b290351.txt">Table of n, a(n) for n = 0..576</a>

%p b:= proc(n) option remember; `if`(n=0, 1, add(

%p b(n-j)*binomial(n-1, j-1), j=1..n))

%p end:

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(d*

%p b(d), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)

%p end:

%p seq(a(n), n=0..30);

%t b[n_]:=b[n]=If[n==0, 1, Sum[b[n - j] Binomial[n - 1, j - 1], {j, n}]]; a[n_]:=a[n]=If[n==0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}] a[n - j], {j, n}]/n]; Table[a[n], {n, 0, 50}] (* _Indranil Ghosh_, Jul 28 2017, after Maple code *)

%Y Cf. A000110, A085686, A290352, A305850.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jul 28 2017