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A327648
Number of parts in all proper many times partitions of n.
4
0, 1, 3, 9, 45, 185, 1277, 7469, 67993, 514841, 5414197, 52609653, 679432169, 7704502013, 111283754969, 1515535050805, 25257251330321, 385282195339393, 7088110874426409, 123325149268482781, 2520808658222616653, 48623257343586890769, 1078165538033926164281
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) = 9 = 1 + 2 + 3 + 3, counting the (final) parts in: 3, 3->21, 3->111, 3->21->111.
a(4) = 45: 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, [1, 0],
`if`(k=0, [1, 1], `if`(i<2, 0, b(n, i-1, k))+
(h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
b(n-i, min(n-i, i), k)))(b(i$2, k-1))))
end:
a:= n-> add(add(b(n$2, i)[2]*(-1)^(k-i)*
binomial(k, i), i=0..k), k=0..n-1):
seq(a(n), n=0..25);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i < 2, 0, b[n, i - 1, k]] + Function[h, Function[f, f + {0, f[[1]] h[[2]]/ h[[1]]}][h[[1]] b[n - i, Min[n - i, i], k]]][b[i, i, k - 1]]]];
a[n_] := Sum[b[n, n, i][[2]] (-1)^(k - i) Binomial[k, i], {k, 0, n - 1}, {i, 0, k}];
a /@ Range[0, 25] (* Jean-François Alcover, May 01 2020, after Maple *)
CROSSREFS
Row sums of A327631.
Sequence in context: A192891 A364296 A068100 * A262129 A012821 A229813
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 20 2019
STATUS
approved