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A271788 Number of set partitions of [n] having exactly one pair (m,m+1) such that m is in some block b and m+1 is in block b+1.

%I A271788 #13 May 27 2018 07:00:54
%S A271788 0,1,3,7,16,39,105,314,1035,3723,14494,60670,271544,1293147,6523495,
%T A271788 34724247,194357190,1140402612,6995760364,44760085240,298054873358,
%U A271788 2061644525813,14787185811993,109804829195145,842928183558160,6680572760715182,54595535222727960
%N A271788 Number of set partitions of [n] having exactly one pair (m,m+1) such that m is in some block b and m+1 is in block b+1.
%H A271788 Alois P. Heinz, <a href="/A271788/b271788.txt">Table of n, a(n) for n = 1..500</a>
%H A271788 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A271788 a(2) = 1: 1|2.
%e A271788 a(3) = 3: 12|3, 13|2, 1|23.
%e A271788 a(4) = 7: 123|4, 124|3, 12|34, 134|2, 13|2|4, 14|23, 1|234.
%e A271788 a(5) = 16: 1234|5, 1235|4, 123|45, 1245|3, 124|3|5, 125|34, 12|345, 1345|2, 134|2|5, 135|2|4, 13|25|4, 13|2|45, 145|23, 14|23|5, 15|234, 1|2345.
%p A271788 b:= proc(n, i, m, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
%p A271788        add(`if`(j=i+1 and k=0, 0, b(n-1, j, max(m, j), k-
%p A271788       `if`(j=i+1, 1, 0))), j=1..m+1))
%p A271788     end:
%p A271788 a:= n-> b(n, 1, 0, 1):
%p A271788 seq(a(n), n=1..30);
%t A271788 b[n_, i_, m_, k_] := b[n, i, m, k] = If[n == 0, If[k == 0, 1, 0], Sum[If[j == i + 1 && k == 0, 0, b[n - 1, j, Max[m, j], k - If[j == i + 1, 1, 0]]], {j, 1, m + 1}]];
%t A271788 a[n_] :=  b[n, 1, 0, 1];
%t A271788 Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, May 27 2018, translated from Maple *)
%Y A271788 Column k=1 of A185982.
%K A271788 nonn
%O A271788 1,3
%A A271788 _Alois P. Heinz_, Apr 14 2016