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A275389 Number of set partitions of [n] with a strongly unimodal block size list.

%I A275389 #12 May 01 2017 16:04:50
%S A275389 1,1,1,4,7,19,71,219,759,2697,12395,47477,231950,1040116,4851742,
%T A275389 26690821,131478031,736418510,4262619682,24680045903,145629814329,
%U A275389 935900941506,5778263418232,37626913475878,257550263109475,1782180357952449,12526035635331581
%N A275389 Number of set partitions of [n] with a strongly unimodal block size list.
%C A275389 Strongly unimodal means strictly increasing then strictly decreasing.
%H A275389 Alois P. Heinz, <a href="/A275389/b275389.txt">Table of n, a(n) for n = 0..677</a>
%H A275389 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A275389 a(3) = 4: 123, 12|3, 13|2, 1|23.
%e A275389 a(4) = 7: 1234, 123|4, 124|3, 134|2, 1|234, 1|23|4, 1|24|3.
%e A275389 a(5) = 19: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 1345|2, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345, 1|234|5, 1|235|4, 1|245|3.
%p A275389 b:= proc(n, i, t) option remember; `if`(t=0 and n>i*(i-1)/2, 0,
%p A275389      `if`(n=0, 1, add(b(n-j, j, 0)*binomial(n-1, j-1), j=1..min(n, i-1))
%p A275389      +`if`(t=1, add(b(n-j, j, 1)*binomial(n-1, j-1), j=i+1..n), 0)))
%p A275389     end:
%p A275389 a:= n-> b(n, 0, 1):
%p A275389 seq(a(n), n=0..30);
%t A275389 b[n_, i_, t_] := b[n, i, t] = If[t==0 && n > i*(i-1)/2, 0, If[n==0, 1, Sum[b[n-j, j, 0]*Binomial[n-1, j-1], {j, 1, Min[n, i-1]}] + If[t==1, Sum[b[n-j, j, 1]*Binomial[n-1, j-1], {j, i+1, n}], 0]]]; a[n_] := b[n, 0, 1]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Feb 07 2017, translated from Maple *)
%Y A275389 Cf. A007837, A038041, A059618, A275309, A275310, A275311, A275312, A275313, A286077.
%K A275389 nonn
%O A275389 0,4
%A A275389 _Alois P. Heinz_, Jul 26 2016