A050344 Number of partitions of n into distinct parts with 3 levels of parentheses.

1, 1, 1, 5, 11, 25, 60, 141, 321, 742, 1688, 3810, 8580, 19225, 42844, 95156, 210480, 463866, 1018957, 2231114, 4870400, 10601805, 23015117, 49833471, 107636878, 231940988, 498671281, 1069826434, 2290402343, 4893782240, 10436263572, 22214850439, 47202869437

Offset: 0

Author

Christian G. Bower, Oct 15 1999

Keywords

• nonn

Examples

			4 = (((4))) = (((3)))+(((1))) = (((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:
f:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(h(i, i), j)*f(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(f(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}]]];
f[n_, i_] := f[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[h[i, i], j]* f[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[f[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, Jun 11 2018, after Alois P. Heinz *)

Formula

Weigh transform of A050343.