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A327647
Number of parts in all proper many times partitions of n into distinct parts.
3
0, 1, 1, 3, 6, 15, 38, 133, 446, 1913, 7492, 36293, 175904, 953729, 5053294, 31353825, 188697696, 1268175779, 8356974190, 61775786301, 448436391810, 3579695446911, 27848806031468, 239229189529685, 2019531300063238, 18477179022470655, 165744369451885256
OFFSET
0,4
COMMENTS
In each step at least one part is replaced by the partition of itself into smaller distinct parts. The parts are not resorted and the parts in the result are not necessarily distinct.
EXAMPLE
a(4) = 6 = 1 + 2 + 3, counting the (final) parts in 4, 4->31, 4->31->211.
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
`if`(k=0, [1, 1], `if`(i*(i+1)/2<n, 0, b(n, i-1, k)+
(h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
b(n-i, min(n-i, i-1), k)))(b(i$2, k-1)))))
end:
a:= n-> add(add(b(n$2, i)[2]*(-1)^(k-i)*
binomial(k, i), i=0..k), k=0..max(0, n-2)):
seq(a(n), n=0..27);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i(i + 1)/2 < n, 0, b[n, i - 1, k] + Function[h, Function[f, f + {0, f[[1]]* h[[2]]/h[[1]]}][h[[1]] b[n-i, Min[n - i, i - 1], k]]][b[i, i, k - 1]]]]];
a[n_] := Sum[Sum[b[n, n, i][[2]] (-1)^(k-i) Binomial[k, i], {i, 0, k}], {k, 0, Max[0, n-2]}];
a /@ Range[0, 27] (* Jean-François Alcover, May 03 2020, after Maple *)
CROSSREFS
Row sums of A327632, A327648.
Sequence in context: A063778 A279374 A087124 * A086326 A098701 A218777
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 20 2019
STATUS
approved