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%I #21 May 10 2021 10:48:48
%S 1,1,1,3,3,5,10,12,17,24,44,51,76,98,138,217,272,366,493,654,848,1284,
%T 1560,2115,2718,3610,4550,6024,8230,10296,13354,17144,21926,27903,
%U 35556,44644,59959,73456,94109,117735,150078,185800,235719,290818,365334,467923
%N Number of ways to choose a set partition of a strict integer partition of n.
%H Alois P. Heinz, <a href="/A294617/b294617.txt">Table of n, a(n) for n = 0..5000</a>
%F A279375(n) <= a(n) <= A279790(n).
%F G.f.: Sum_{k>=0} Bell(k) * x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j). - _Ilya Gutkovskiy_, Jan 28 2020
%e The a(6) = 10 set partitions are: {{6}}, {{1},{5}}, {{5,1}}, {{2},{4}}, {{4,2}}, {{1},{2},{3}}, {{1},{3,2}}, {{2,1},{3}}, {{3,1},{2}}, {{3,2,1}}.
%p b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
%p `if`(n=0, combinat[bell](t), b(n, i-1, t)+
%p `if`(i>n, 0, b(n-i, min(n-i, i-1), t+1))))
%p end:
%p a:= n-> b(n$2, 0):
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Nov 07 2017
%t Table[Total[BellB[Length[#]]&/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,25}]
%t (* Second program: *)
%t b[n_, i_, t_] := b[n, i, t] = If[n > i (i + 1)/2, 0, If[n == 0, BellB[t], b[n, i - 1, t] + If[i > n, 0, b[n - i, Min[n - i, i - 1], t + 1]]]];
%t a[n_] := b[n, n, 0];
%t a /@ Range[0, 50] (* _Jean-François Alcover_, May 10 2021, after _Alois P. Heinz_ *)
%Y Cf. A000009, A000110, A063834, A258466, A259936, A279375, A279785, A279790.
%Y Row sums of A330460 and of A330759.
%K nonn
%O 0,4
%A _Gus Wiseman_, Nov 05 2017
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