A261489 Number of partitions of subsets of {1,...,n}, where consecutive integers and the elements in {1, n} are required to be in different parts.

1, 2, 4, 8, 25, 82, 313, 1318, 6098, 30603, 165282, 954065, 5853242, 37987146, 259751877, 1864926846, 14016442573, 109985575616, 898948324164, 7637000950875, 67310106587314, 614420757079213, 5799709014601124, 56530981389520624, 568255134674637557

Offset: 0

Author

Alois P. Heinz, Aug 21 2015

Keywords

• nonn

Examples

			a(3) = 8: {}, 1, 2, 3, 1|2, 1|3, 2|3, 1|2|3.
a(4) = 25: {}, 1, 2, 3, 4, 1|2, 1|3, 13, 1|4, 2|3, 2|4, 24, 3|4, 1|2|3, 13|2, 1|2|4, 1|24, 1|3|4, 13|4, 2|3|4, 24|3, 1|2|3|4, 13|2|4, 1|3|24, 13|24.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A247100, A261134, A261041, A261492.

Programs

Maple

g:= proc(n, l, t, f) option remember; `if`(n=0, 1,
      add(`if`(l>0 and j=l or f=1 and n=1 and j=1, 0,
      g(n-1, j, t+`if`(j=t, 1, 0), f)), j=0..t))
    end:
a:= n-> `if`(n=0, 1, g(n-1, 0, 1, 0)+g(n-1, 1, 2, 1)):
seq(a(n), n=0..25);

Mathematica

g[n_, l_, t_, f_] := g[n, l, t, f] = If[n==0, 1, Sum[If[l>0 && j==l || f==1 && n==1 && j==1, 0, g[n-1, j, t+If[j==t, 1, 0], f]], {j, 0, t}]]; a[n_] := If[n==0, 1, g[n-1, 0, 1, 0]+g[n-1, 1, 2, 1]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 02 2017, translated from Maple *)