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.

1, 6, 24, 91, 374, 1699, 8410, 44794, 254718, 1538027, 9818858, 66030017, 466215802, 3446197857, 26600048069, 213901723087, 1788292021799, 15514751860549, 139443578638933, 1296371888068649, 12448726758061263, 123316489529161713, 1258632265803403708

Offset: 3

Author

Alois P. Heinz, Mar 26 2016

Keywords

• nonn

Examples

			a(3) = 1: 13|2.
a(4) = 6: 124|3, 134|2, 13|24, 13|2|4, 14|23, 1|24|3.
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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..250
• Wikipedia, Partition of a set

Crossrefs

Column k=1 of A270953.

Programs

Maple

b:= proc(n, i, m) option remember; convert(series(
      `if`(n=0, 1, add(b(n-1, j, max(m, j))*
      `if`(j=i-1, x, 1), j=1..m+1)), x, 2), polynom)
    end:
a:= n-> coeff(b(n, 1, 0), x, 1):
seq(a(n), n=3..30);

Mathematica

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;
a[n_] := Coefficient[b[n, 1, 0], x, 1];
Table[a[n], {n, 3, 30}] (* Jean-François Alcover, May 27 2018, translated from Maple *)