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

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

N. J. A. Sloane, Transforms

Formula

Weigh transform of A050343.

Example

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

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 *)

Crossrefs

Cf. A050342-A050350.

Sequence in context: A014858 A355243 A018368 * A318033 A326161 A354838

Adjacent sequences: A050341 A050342 A050343 * A050345 A050346 A050347

Keyword

nonn

Author

Christian G. Bower, Oct 15 1999

Status

approved