A327646 Total number of steps in all proper many times partitions of n.

0, 0, 1, 4, 25, 108, 788, 4740, 44445, 339632, 3625136, 35508536, 462626736, 5273725108, 76634997096, 1047347436984, 17542238923677, 268193251446228, 4949536256552648, 86303019303031400, 1768833677916545596, 34165810747993948664, 759192269597947084836

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) = 4 = 0+1+1+2, counting steps "->" in: 3, 3->21, 3->111, 3->21->111.
a(4) = 25: 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..400

Crossrefs

Cf. A327639.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
      b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
a:= n-> add(k*add(b(n$2, i)*(-1)^(k-i)*
        binomial(k, i), i=0..k), k=1..n-1):
seq(a(n), n=0..23);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0, 1, If[i > 1, b[n, i - 1, k], 0] + b[i, i, k - 1] b[n - i, Min[n - i, i], k]];
a[n_] := Sum[k Sum[b[n, n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}], {k, 1, n - 1}];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=1..n-1} k * A327639(n,k).