A327623 Number of parts in all n-times partitions of n into distinct parts.

0, 1, 1, 7, 27, 121, 553, 3865, 24625, 202954, 1519540, 14193455, 132441998, 1381539355, 14096067555, 168745220585, 1961128020387, 25473872598375, 324797436024684, 4647784901400988, 65394584337577858, 1012005650484163962, 15285115573675197704

Offset: 0

Author

Alois P. Heinz, Sep 19 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300
• Wikipedia, Partition (number theory)

Crossrefs

Main diagonal of A327622.

Cf. A327619.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
     `if`(k=0, [1, 1], `if`(i*(i+1)/2 (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
        b(n-i, min(n-i, i-1), k)))(b(i$2, k-1)))))
    end:
a:= n-> b(n$3)[2]:
seq(a(n), n=0..23);

Mathematica

b[n_, i_, k_] := b[n, i, k] = With[{}, If[n == 0, Return[{1, 0}]]; If[k == 0, Return[{1, 1}]]; If[i (i + 1)/2 < n, Return[{0, 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 - 1], k]]][b[i, i, k - 1]]];
a[n_] := b[n, n, n][[2]];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)