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%I #10 May 04 2020 07:04:44
%S 0,0,0,1,4,10,60,35,336,1848,16080,33825,93280,539396,3856216,
%T 49390250,147478800,708041160,2354289744,18196716309,150847235040,
%U 2615953578700,9488756856040,57565330671310,296745669669768,1435526275752900,12231628020365000
%N Number of set partitions of [n] with distinct block sizes and one of the block sizes is 3.
%H Alois P. Heinz, <a href="/A328155/b328155.txt">Table of n, a(n) for n = 0..697</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%p b:= proc(n, i, k) option remember; `if`(i*(i+1)/2<n, 0,
%p `if`(n=0, 1, `if`(i<2, 0, b(n, i-1, `if`(i=k, 0, k)))+
%p `if`(i=k, 0, b(n-i, min(n-i, i-1), k)*binomial(n, i))))
%p end:
%p a:= n-> b(n$2, 0)-b(n$2, 3):
%p seq(a(n), n=0..29);
%t b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, If[n == 0, 1, If[i < 2, 0, b[n, i - 1, If[i == k, 0, k]]] + If[i == k, 0, b[n - i, Min[n - i, i - 1], k]*Binomial[n, i]]]];
%t a[n_] := b[n, n, 0] - b[n, n, 3];
%t a /@ Range[0, 29] (* _Jean-François Alcover_, May 04 2020, after Maple *)
%Y Column k=3 of A327869.
%Y Cf. A328153.
%K nonn
%O 0,5
%A _Alois P. Heinz_, Oct 05 2019
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