A271788 - OEIS

%I #13 May 27 2018 07:00:54

%S 0,1,3,7,16,39,105,314,1035,3723,14494,60670,271544,1293147,6523495,

%T 34724247,194357190,1140402612,6995760364,44760085240,298054873358,

%U 2061644525813,14787185811993,109804829195145,842928183558160,6680572760715182,54595535222727960

%N 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 Alois P. Heinz, <a href="/A271788/b271788.txt">Table of n, a(n) for n = 1..500</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%e a(2) = 1: 1|2.

%e a(3) = 3: 12|3, 13|2, 1|23.

%e a(4) = 7: 123|4, 124|3, 12|34, 134|2, 13|2|4, 14|23, 1|234.

%e 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 b:= proc(n, i, m, k) option remember; `if`(n=0, `if`(k=0, 1, 0),

%p        add(`if`(j=i+1 and k=0, 0, b(n-1, j, max(m, j), k-

%p       `if`(j=i+1, 1, 0))), j=1..m+1))

%p     end:

%p a:= n-> b(n, 1, 0, 1):

%p seq(a(n), n=1..30);

%t 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 a[n_] :=  b[n, 1, 0, 1];

%t Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, May 27 2018, translated from Maple *)

%Y Column k=1 of A185982.

%K nonn

%O 1,3

%A _Alois P. Heinz_, Apr 14 2016