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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 20 2019
STATUS
approved