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A327644
Number of proper many times partitions of n.
3
1, 1, 2, 4, 14, 44, 244, 1196, 9366, 62296, 584016, 5120548, 60244028, 627389924, 8378159376, 106097674780, 1652301306958, 23655318730276, 409987534384504, 6742903763089068, 130675390985884516, 2396246933608687036, 50636625943991790784, 1032841246318579471748
OFFSET
0,3
COMMENTS
In each step at least one part is replaced by the partition of itself into smaller parts. The parts are not resorted.
EXAMPLE
a(3) = 4: 3, 3->21, 3->111, 3->21->111.
a(4) = 14: 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(add(b(n$2, i)*(-1)^(k-i)*
binomial(k, i), i=0..k), k=0..max(0, 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[b[n, n, i] (-1)^(k - i) Binomial[k, i], {k, 0, Max[0, n - 1]}, {i, 0, k}];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)
CROSSREFS
Row sums of A327639.
Cf. A327648.
Sequence in context: A007866 A226909 A121751 * A151355 A014272 A070822
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 20 2019
STATUS
approved