OFFSET
0,2
COMMENTS
From Gus Wiseman, Nov 25 2019: (Start)
Conjecture: Also the number of set partitions of {1, ..., n + 1} where, if x and x + 2 belong to the same block, then so does x + 1. For example, the a(0) = 1 through a(3) = 10 set partitions are:
{{1}} {{1,2}} {{1,2,3}} {{1,2,3,4}}
{{1},{2}} {{1},{2,3}} {{1},{2,3,4}}
{{1,2},{3}} {{1,2},{3,4}}
{{1},{2},{3}} {{1,2,3},{4}}
{{1,4},{2,3}}
{{1},{2},{3,4}}
{{1},{2,3},{4}}
{{1,2},{3},{4}}
{{1,4},{2},{3}}
{{1},{2},{3},{4}}
(End)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..576
Notamathematician et al., Generating function for A261041, MathOverflow, 2024.
FORMULA
From Max Alekseyev, Sep 08 2024: (Start)
a(n) = Sum_{k=0..n} A000110(k) * Sum_{j=0..[(n-k)/2]} binomial(k+j-1,j).
G.f.: 1/(1-x) * Sum_{k>=0} A000110(k) * (x/(1-x^2))^k. (End)
EXAMPLE
For n=3 the a(3) = 10 partitions are {}, 1, 2, 3, 1|2, 13, 1|3, 2|3, 13|2, 1|2|3.
From Gus Wiseman, Nov 25 2019: (Start)
The a(0) = 1 through a(3) = 10 set partitions:
{} {} {} {}
{{1}} {{1}} {{1}}
{{2}} {{2}}
{{1},{2}} {{3}}
{{1,3}}
{{1},{2}}
{{1},{3}}
{{2},{3}}
{{1,3},{2}}
{{1},{2},{3}}
(End)
MAPLE
g:= proc(n, l, t) option remember; `if`(n=0, 1, add(`if`(l>0
and j=l, 0, g(n-1, j, `if`(j=t, t+1, t))), j=0..t))
end:
a:= n-> g(n, 0, 1):
seq(a(n), n=0..30);
MATHEMATICA
g[n_, l_, t_] := g[n, l, t] = If[n==0, 1, Sum[If[l>0 && j==l, 0, g[n-1, j, If[j==t, t+1, t]]], {j, 0, t}]]; a[n_] := g[n, 0, 1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 04 2017, translated from Maple *)
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
Table[Length[Select[Join@@sps/@Subsets[Range[n]], !MemberQ[#, {___, x_, y_, ___}/; x+1==y]&]], {n, 0, 6}] (* Gus Wiseman, Nov 25 2019 *)
PROG
(PARI) a261041(n) = sum(k=0, n, sum(j=0, k, stirling(k, j, 2)) * sum(j=0, (n-k)\2, binomial(k+j-1, j))); \\ Max Alekseyev, Sep 08 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 09 2015
STATUS
approved