A050343 Number of partitions of n into distinct parts with 2 levels of parentheses.

1, 1, 1, 4, 7, 14, 29, 57, 110, 217, 417, 794, 1513, 2860, 5373, 10063, 18740, 34750, 64221, 118199, 216775, 396297, 722136, 1311888, 2376575, 4293407, 7735941, 13903985, 24929763, 44595606, 79598328, 141770576, 251984463, 446991405, 791391545, 1398551523

Offset: 0

Author

Christian G. Bower, Oct 15 1999

Keywords

• nonn

Examples

			4 = ((4)) = ((3))+((1)) = ((3)+(1)) = ((3+1)) = ((2+1))+((1)) = ((2+1)+(1)).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A050342-A050350.

Programs

Maple

g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      g(n, i-1)+`if`(i>n, 0, g(n-i, i-1))))
    end:
h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i, i), j)*h(n-i*j, i-1), j=0..n/i)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(h(i, i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..50); # Alois P. Heinz, May 19 2013

Mathematica

g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, g[n, i-1] + If[i>n, 0, g[n-i, i-1]]]] ; h[n_, i_] := h[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[g[i, i], j]*h[n-i*j, i-1], {j, 0, n/i}]]]; b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[ Binomial[ h[i, i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 17 2015, after Alois P. Heinz *)

Formula

Weigh transform of A050342.