OFFSET
1,3
COMMENTS
In other words, if the smallest block in a partition has size k then there are no other blocks in the partition with size k.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..576
FORMULA
E.g.f.: Sum_{k>=1} x^k/k! * exp(exp(x) - Sum_{i=0..k} x^i/i!).
EXAMPLE
a(4) = 5 because we have: {{1,2,3,4}}, {{1},{2,3,4}}, {{1,3,4},{2}}, {{1,2,3},{4}}, {{1,2,4},{3}}.
MAPLE
with(combinat):
b:= proc(n, i) option remember;
`if`(i<1, 0, `if`(n=i, 1, 0)+add(b(n-i*j, i-1)*
multinomial(n, n-i*j, i$j)/j!, j=0..(n-1)/i))
end:
a:= n-> b(n$2):
seq(a(n), n=1..25); # Alois P. Heinz, Jul 07 2016
MATHEMATICA
nn=25; Drop[Range[0, nn]!CoefficientList[Series[Sum[x^k/k!Exp[Exp[x]-Sum[x^i/i!, {i, 0, k}]], {k, 1, nn}], {x, 0, nn}], x], 1]
(* Second program: *)
multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[i<1, 0, If[n==i, 1, 0] + Sum[b[n-i*j, i-1]*multinomial[n, Prepend[Array[i&, j], n-i*j]]/j!, {j, 0, (n-1)/i}]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Feb 03 2017, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Apr 01 2013
STATUS
approved