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

0,4

Comments

In each step at least one part is replaced by the partition of itself into smaller parts. The parts are not resorted.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..400

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A327639.

Sequence in context: A225692 A359524 A070764 * A244746 A110051 A334551

Adjacent sequences: A327643 A327644 A327645 * A327647 A327648 A327649

Keyword

nonn

Author

Alois P. Heinz, Sep 20 2019

Status

approved