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A270955 Number of set partitions of [n] having exactly one pair (m,m+1) such that m+1 is in some block b and m is in block b+1.

%I A270955 #9 May 27 2018 07:00:10
%S A270955 1,6,24,91,374,1699,8410,44794,254718,1538027,9818858,66030017,
%T A270955 466215802,3446197857,26600048069,213901723087,1788292021799,
%U A270955 15514751860549,139443578638933,1296371888068649,12448726758061263,123316489529161713,1258632265803403708
%N A270955 Number of set partitions of [n] having exactly one pair (m,m+1) such that m+1 is in some block b and m is in block b+1.
%H A270955 Alois P. Heinz, <a href="/A270955/b270955.txt">Table of n, a(n) for n = 3..250</a>
%H A270955 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A270955 a(3) = 1: 13|2.
%e A270955 a(4) = 6: 124|3, 134|2, 13|24, 13|2|4, 14|23, 1|24|3.
%e A270955 a(5) = 24: 1235|4, 1245|3, 124|35, 124|3|5, 125|34, 12|35|4, 1345|2, 134|25, 134|2|5, 13|245, 13|24|5, 135|2|4, 13|2|45, 13|2|4|5, 145|23, 14|235, 14|23|5, 15|234, 1|235|4, 1|245|3, 1|24|35, 1|24|3|5, 1|25|34, 1|2|35|4.
%p A270955 b:= proc(n, i, m) option remember; convert(series(
%p A270955       `if`(n=0, 1, add(b(n-1, j, max(m, j))*
%p A270955       `if`(j=i-1, x, 1), j=1..m+1)), x, 2), polynom)
%p A270955     end:
%p A270955 a:= n-> coeff(b(n, 1, 0), x, 1):
%p A270955 seq(a(n), n=3..30);
%t A270955 b[n_, i_, m_] := b[n, i, m] = If[n == 0, 1, Sum[b[n - 1, j, Max[m, j]]*If[j == i - 1, x, 1], {j, 1, m + 1}]] + O[x]^2 // Normal;
%t A270955 a[n_] := Coefficient[b[n, 1, 0], x, 1];
%t A270955 Table[a[n], {n, 3, 30}] (* _Jean-François Alcover_, May 27 2018, translated from Maple *)
%Y A270955 Column k=1 of A270953.
%K A270955 nonn
%O A270955 3,2
%A A270955 _Alois P. Heinz_, Mar 26 2016