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A271763 Number of set partitions of [n] with minimal block length multiplicity equal to three.

%I A271763 #11 May 15 2018 06:41:22
%S A271763 1,0,0,15,0,0,1540,3150,24255,81235,496210,605605,36987951,13833820,
%T A271763 849333940,24419945732,111237098546,1219799147204,16146398449224,
%U A271763 109697049177254,1037441240056529,9042707959752775,84237798887033660,614681985047225810
%N A271763 Number of set partitions of [n] with minimal block length multiplicity equal to three.
%H A271763 Alois P. Heinz, <a href="/A271763/b271763.txt">Table of n, a(n) for n = 3..576</a>
%H A271763 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A271763 a(n) = A271424(n,3).
%e A271763 a(6) = 15: 12|34|56, 12|35|46, 12|36|45, 13|24|56, 13|25|46, 13|26|45, 14|23|56, 15|23|46, 16|23|45, 14|25|36, 14|26|35, 15|24|36, 16|24|35, 15|26|34, 16|25|34.
%p A271763 with(combinat):
%p A271763 b:= proc(n, i, k) option remember; `if`(n=0, 1,
%p A271763       `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
%p A271763         *b(n-i*j, i-1, k)/j!, j={0, $k..n/i})))
%p A271763     end:
%p A271763 a:= n-> b(n$2, 3)-b(n$2, 4):
%p A271763 seq(a(n), n=3..30);
%t A271763 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A271763 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 A271763 a[n_] := b[n, n, 3] - b[n, n, 4];
%t A271763 Table[a[n], {n, 3, 30}] (* _Jean-François Alcover_, May 15 2018, after _Alois P. Heinz_ *)
%Y A271763 Column k=3 of A271424.
%K A271763 nonn
%O A271763 3,4
%A A271763 _Alois P. Heinz_, Apr 13 2016