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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 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..300

Wikipedia, Partition (number theory)

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-Fran├žois Alcover, May 01 2020, after Maple *)

CROSSREFS

Row sums of A327631.

Cf. A327644, A327647.

Sequence in context: A352797 A192891 A068100 * A262129 A012821 A229813

Adjacent sequences:  A327645 A327646 A327647 * A327649 A327650 A327651

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Sep 20 2019

STATUS

approved

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Last modified September 26 12:17 EDT 2022. Contains 356997 sequences. (Running on oeis4.)