A327623 - OEIS

%I #12 Dec 09 2020 10:56:42

%S 0,1,1,7,27,121,553,3865,24625,202954,1519540,14193455,132441998,

%T 1381539355,14096067555,168745220585,1961128020387,25473872598375,

%U 324797436024684,4647784901400988,65394584337577858,1012005650484163962,15285115573675197704

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

%H Alois P. Heinz, <a href="/A327623/b327623.txt">Table of n, a(n) for n = 0..300</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>

%p b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],

%p      `if`(k=0, [1, 1], `if`(i*(i+1)/2<n, 0, b(n, i-1, k)+

%p          (h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*

%p         b(n-i, min(n-i, i-1), k)))(b(i$2, k-1)))))

%p     end:

%p a:= n-> b(n$3)[2]:

%p seq(a(n), n=0..23);

%t 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]]];

%t a[n_] := b[n, n, n][[2]];

%t a /@ Range[0, 23] (* _Jean-François Alcover_, Dec 09 2020, after _Alois P. Heinz_ *)

%Y Main diagonal of A327622.

%Y Cf. A327619.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Sep 19 2019