A224219 Number of set partitions of {1,2,...,n} such that the size of the smallest block is unique.

1, 1, 4, 5, 31, 82, 344, 1661, 7942, 38721, 228680, 1377026, 8529756, 56756260, 402300799, 2960135917, 22692746719, 181667760724, 1516381486766, 13135566948285, 117868982320877, 1093961278908818, 10492653292100919, 103880022098900234, 1059925027073166856

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

Cf. A224244, A372721, A372802.

Column k=1 of A372762.

Sequence in context: A124482 A265709 A265708 * A128867 A262031 A326998

Adjacent sequences: A224216 A224217 A224218 * A224220 A224221 A224222

Keyword

nonn

Author

Geoffrey Critzer, Apr 01 2013

Status

approved