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

0,4

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.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..300

Wikipedia, Partition (number theory)

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

Adjacent sequences: A327644 A327645 A327646 * A327648 A327649 A327650

Keyword

nonn

Author

Alois P. Heinz, Sep 20 2019

Status

approved