A271426 - OEIS

%I #10 May 07 2018 09:49:43

%S 0,1,1,4,11,51,132,771,3089,18388,96423,627529,3349018,24510305,

%T 155908651,1171494200,8647906143,71603237483,572103586280,

%U 5172888505403,43344865682187,416735802793600,3830340992280773,38239507035358011,374336654847685014

%N Number of set partitions of [n] with minimal block length multiplicity equal to one.

%C At least one block length occurs exactly once.

%H Alois P. Heinz, <a href="/A271426/b271426.txt">Table of n, a(n) for n = 0..576</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%F a(n) = A271424(n,1).

%e a(1) = 1: 1.

%e a(2) = 1: 12.

%e a(3) = 4: 123, 12|3, 13|2, 1|23.

%e a(4) = 11: 1234, 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34.

%p with(combinat):

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

%p       `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)

%p         *b(n-i*j, i-1, k)/j!, j={0,$k..n/i})))

%p     end:

%p a:= n-> b(n$2, 1)-b(n$2, 2):

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

%t multinomial[n_, k_List] := n!/Times @@ (k!);

%t b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Join[{n - i*j}, Table[i, j]]]*b[n - i*j, i - 1, k]/j!, {j, Join[{0}, Range[k, n/i]]}]]];

%t a[n_] := b[n, n, 1] - b[n, n, 2];

%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, May 07 2018, after _Alois P. Heinz_ *)

%Y Column k=1 of A271424.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Apr 07 2016