A144792 EXP transform of A140585.

1, 1, 5, 33, 282, 2938, 36029, 507440, 8058990, 142315830, 2763775025, 58498072273, 1339545500214, 32980132065364, 868417100538399, 24344702489881998, 723694354351500431, 22733368105181643193, 752291980101845144878, 26153153055424960528533

Offset: 0

Author

Thomas Wieder, Sep 21 2008

Keywords

• nonn

Comments

Stirling transform of A143463.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A140585, A143463.

Programs

Maple

with(numtheory): with(combinat): b:= proc(k) option remember; add(d/d!^(k/d), d=divisors(k)) end: c:= proc(n) option remember; `if`(n=0, 1, add((n-1)!/(n-k)!* b(k)* c(n-k), k=1..n)) end: aa:= n-> add(stirling2(n, k) *c(k), k=1..n): a:= proc(n) option remember; `if`(n=0, 1, aa(n)+ add(binomial(n-1, k-1) *aa(k) *a(n-k), k=1..n-1)) end: seq(a(n), n=1..20); # Alois P. Heinz, Oct 10 2008

Mathematica

b[k_] := b[k] = DivisorSum[k, #/#!^(k/#)&]; c[n_] := c[n] = If[n==0, 1, Sum[(n-1)!/(n-k)!*b[k]*c[n-k], {k, 1, n}]]; aa[n_] := Sum[StirlingS2[n, k]*c[k], {k, 1, n}]; a[n_] := a[n] = If[n==0, 1, aa[n] + Sum[Binomial[ n-1, k-1]*aa[k]*a[n-k], {k, 1, n-1}]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 25 2017, after Alois P. Heinz *)

Formula

E.g.f: (1/exp(1)) exp( 1 / prod_{k=1}^{inf} (1 - (exp(x)-1)^k / k!) ).

a(n) = sum_{k=1..n} C(n-1,k-1) A140585(k) a(n-k).

With S2(n,k) as the Stirling number of the second kind we have

a(n) = sum_{k=1..n} A143463(n) S2(n,k).

Extensions

More terms from Alois P. Heinz, Oct 10 2008