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A327646
Total number of steps in all proper many times partitions of n.
2
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
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 20 2019
STATUS
approved