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A224219 Number of set partitions of {1,2,...,n} such that the size of the smallest block is unique. 4
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Feb 03 2017, after Alois P. Heinz *)

CROSSREFS

Cf. A224244.

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

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Last modified January 18 07:55 EST 2022. Contains 350454 sequences. (Running on oeis4.)