%I #16 Dec 19 2015 14:18:57

%S 1,1,1,4,7,14,29,57,110,217,417,794,1513,2860,5373,10063,18740,34750,

%T 64221,118199,216775,396297,722136,1311888,2376575,4293407,7735941,

%U 13903985,24929763,44595606,79598328,141770576,251984463,446991405,791391545,1398551523

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

%H Alois P. Heinz, <a href="/A050343/b050343.txt">Table of n, a(n) for n = 0..1000</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F Weigh transform of A050342.

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

%p g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p g(n, i-1)+`if`(i>n, 0, g(n-i, i-1))))

%p end:

%p h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p add(binomial(g(i, i), j)*h(n-i*j, i-1), j=0..n/i)))

%p end:

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p add(binomial(h(i, i), j)*b(n-i*j, i-1), j=0..n/i)))

%p end:

%p a:= n-> b(n, n):

%p seq(a(n), n=0..50); # _Alois P. Heinz_, May 19 2013

%t 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_ *)

%Y Cf. A050342-A050350.

%K nonn

%O 0,4

%A _Christian G. Bower_, Oct 15 1999