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A275313 Number of set partitions of [n] where adjacent blocks differ in size.

%I A275313 #17 May 01 2017 14:55:30
%S A275313 1,1,1,4,7,23,100,333,1443,6910,36035,186958,1095251,6620976,42151463,
%T A275313 290483173,2030271491,15044953241,116044969497,930056879535,
%U A275313 7749440529803,66931578540965,597728811956244,5511695171795434,52578231393128128,515775207055816041
%N A275313 Number of set partitions of [n] where adjacent blocks differ in size.
%H A275313 Alois P. Heinz, <a href="/A275313/b275313.txt">Table of n, a(n) for n = 0..580</a>
%H A275313 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A275313 a(3) = 4: 123, 12|3, 13|2, 1|23.
%e A275313 a(4) = 7: 1234, 123|4, 124|3, 134|2, 1|234, 1|23|4, 1|24|3.
%e A275313 a(5) = 23: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|3|45, 1345|2, 134|25, 135|24, 13|245, 13|2|45, 145|23, 14|235, 15|234, 1|2345, 1|234|5, 1|235|4, 14|2|35, 1|245|3, 15|2|34.
%p A275313 b:= proc(n, i) option remember; `if`(n=0, 1, add(`if`(i=j, 0,
%p A275313       b(n-j, `if`(j>n-j, 0, j))*binomial(n-1, j-1)), j=1..n))
%p A275313     end:
%p A275313 a:= n-> b(n, 0):
%p A275313 seq(a(n), n=0..35);
%t A275313 b[n_, i_] := b[n, i] = If[n==0, 1, Sum[If[i==j, 0, b[n-j, If[j>n-j, 0, j]]* Binomial[n-1, j-1]], {j, 1, n}]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Dec 18 2016, after _Alois P. Heinz_ *)
%Y A275313 Cf. A007837, A038041, A275309, A275310, A275311, A275312, A275679, A280275, A286072.
%K A275313 nonn
%O A275313 0,4
%A A275313 _Alois P. Heinz_, Jul 22 2016