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

%I A050344 #17 Jun 11 2018 08:50:38
%S A050344 1,1,1,5,11,25,60,141,321,742,1688,3810,8580,19225,42844,95156,210480,
%T A050344 463866,1018957,2231114,4870400,10601805,23015117,49833471,107636878,
%U A050344 231940988,498671281,1069826434,2290402343,4893782240,10436263572,22214850439,47202869437
%N A050344 Number of partitions of n into distinct parts with 3 levels of parentheses.
%H A050344 Alois P. Heinz, <a href="/A050344/b050344.txt">Table of n, a(n) for n = 0..1000</a>
%H A050344 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A050344 Weigh transform of A050343.
%e A050344 4 = (((4))) = (((3)))+(((1))) = (((3))+((1))) = ((3)+(1)) = ((3+1)) = ((2+1))+((1)) = ((2+1)+(1)).
%p A050344 g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A050344       g(n, i-1)+`if`(i>n, 0, g(n-i, i-1))))
%p A050344     end:
%p A050344 h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A050344       add(binomial(g(i, i), j)*h(n-i*j, i-1), j=0..n/i)))
%p A050344     end:
%p A050344 f:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A050344       add(binomial(h(i, i), j)*f(n-i*j, i-1), j=0..n/i)))
%p A050344     end:
%p A050344 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A050344       add(binomial(f(i, i), j)*b(n-i*j, i-1), j=0..n/i)))
%p A050344     end:
%p A050344 a:= n-> b(n, n):
%p A050344 seq(a(n), n=0..50);  # _Alois P. Heinz_, May 19 2013
%t A050344 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]]]];
%t A050344 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}]]];
%t A050344 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}]]];
%t A050344 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}]]];
%t A050344 a[n_] := b[n, n];
%t A050344 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jun 11 2018, after _Alois P. Heinz_ *)
%Y A050344 Cf. A050342-A050350.
%K A050344 nonn
%O A050344 0,4
%A A050344 _Christian G. Bower_, Oct 15 1999