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A050343
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Number of partitions of n into distinct parts with 2 levels of parentheses.
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22
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1, 1, 1, 4, 7, 14, 29, 57, 110, 217, 417, 794, 1513, 2860, 5373, 10063, 18740, 34750, 64221, 118199, 216775, 396297, 722136, 1311888, 2376575, 4293407, 7735941, 13903985, 24929763, 44595606, 79598328, 141770576, 251984463, 446991405, 791391545, 1398551523
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OFFSET
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0,4
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LINKS
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FORMULA
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EXAMPLE
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4 = ((4)) = ((3))+((1)) = ((3)+(1)) = ((3+1)) = ((2+1))+((1)) = ((2+1)+(1)).
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MAPLE
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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:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(h(i, i), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> b(n, n):
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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