A327648 Number of parts in all proper many times partitions of n.

0, 1, 3, 9, 45, 185, 1277, 7469, 67993, 514841, 5414197, 52609653, 679432169, 7704502013, 111283754969, 1515535050805, 25257251330321, 385282195339393, 7088110874426409, 123325149268482781, 2520808658222616653, 48623257343586890769, 1078165538033926164281

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 parts. The parts are not resorted.

Examples

			a(3) = 9 = 1 + 2 + 3 + 3, counting the (final) parts in: 3, 3->21, 3->111, 3->21->111.
a(4) = 45: 4, 4->31, 4->22, 4->211, 4->1111, 4->31->211, 4->31->1111, 4->22->112, 4->22->211, 4->22->1111, 4->211->1111, 4->31->211->1111, 4->22->112->1111, 4->22->211->1111.
		

Links

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

Crossrefs

Row sums of A327631.

Cf. A327644, A327647.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
     `if`(k=0, [1, 1], `if`(i<2, 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), 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..n-1):
seq(a(n), n=0..25);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i < 2, 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], k]]][b[i, i, k - 1]]]];
a[n_] := Sum[b[n, n, i][[2]] (-1)^(k - i) Binomial[k, i], {k, 0, n - 1}, {i, 0, k}];
a /@ Range[0, 25] (* Jean-François Alcover, May 01 2020, after Maple *)