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A050344 Number of partitions of n into distinct parts with 3 levels of parentheses. 2
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Jun 11 2018, after Alois P. Heinz *)

CROSSREFS

Cf. A050342-A050350.

Sequence in context: A003598 A014858 A018368 * A318033 A326161 A038253

Adjacent sequences:  A050341 A050342 A050343 * A050345 A050346 A050347

KEYWORD

nonn

AUTHOR

Christian G. Bower, Oct 15 1999

STATUS

approved

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Last modified December 15 20:00 EST 2019. Contains 330000 sequences. (Running on oeis4.)