A270703 Total sum of the sizes of all blocks with maximal element n in all set partitions of {1,2,...,2n-1}.

1, 4, 41, 670, 15717, 492112, 19610565, 961547874, 56562256041, 3914022281500, 313638627550657, 28730918805512678, 2976543225606178893, 345587228510915829224, 44615408909143456529309, 6361213086726610526079402, 995709801367376369056571089

Offset: 1

Author

Alois P. Heinz, Mar 21 2016

Keywords

• nonn

Comments

Also total sum of the sizes of all blocks with minimal element n in all set partitions of {1,2,...,2n-1}.

Examples

			a(2) = 4 = 0+2+1+0+1 = sum of the sizes of all blocks with maximal element 2 in all set partitions of {1,2,3}: 123, 12|3, 13|2, 1|23, 1|2|3.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200
• Wikipedia, Partition of a set

Crossrefs

Cf. A000110, A270701, A270702.

Programs

Maple

b:= proc(n, m, t) option remember; `if`(n=0, [1, 0], add(
     `if`(t=1 and j<>m+1, 0, (p->p+`if`(j=-t or t=1 and j=m+1,
      [0, p[1]], 0))(b(n-1, max(m, j), `if`(t=1 and j=m+1, -j,
     `if`(t<0, t, `if`(t>0, t-1, 0)))))), j=1..m+1))
    end:
a:= n-> b(2*n-1, 0, n)[2]:
seq(a(n), n=1..20);

Mathematica

b[n_, m_, t_] := b[n, m, t] = If[n==0, {1, 0}, Sum[If[t==1 && j != m+1, 0, Function[p, p+If[j == -t || t == 1 && j == m+1, {0, p[[1]]}, 0]][b[n-1, Max[m, j], If[t == 1 && j == m+1, -j, If[t<0, t, If[t>0, t-1, 0]]]]]], {j, 1, m+1}]]; a[n_] := b[2*n-1, 0, n][[2]]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Feb 15 2017, translated from Maple *)

Formula

a(n) = A270701(2n-1,n) = A270702(2n-1,n).