A327647 Number of parts in all proper many times partitions of n into distinct parts.

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

Author

Alois P. Heinz, Sep 20 2019

Keywords

• nonn

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.

Examples

			a(4) = 6 = 1 + 2 + 3, counting the (final) parts in 4, 4->31, 4->31->211.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300
• Wikipedia, Partition (number theory)

Crossrefs

Row sums of A327632, A327648.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
     `if`(k=0, [1, 1], `if`(i*(i+1)/2 (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 *)