cp's OEIS Frontend

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

%I A327623 #12 Dec 09 2020 10:56:42
%S A327623 0,1,1,7,27,121,553,3865,24625,202954,1519540,14193455,132441998,
%T A327623 1381539355,14096067555,168745220585,1961128020387,25473872598375,
%U A327623 324797436024684,4647784901400988,65394584337577858,1012005650484163962,15285115573675197704
%N A327623 Number of parts in all n-times partitions of n into distinct parts.
%H A327623 Alois P. Heinz, <a href="/A327623/b327623.txt">Table of n, a(n) for n = 0..300</a>
%H A327623 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%p A327623 b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
%p A327623      `if`(k=0, [1, 1], `if`(i*(i+1)/2<n, 0, b(n, i-1, k)+
%p A327623          (h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
%p A327623         b(n-i, min(n-i, i-1), k)))(b(i$2, k-1)))))
%p A327623     end:
%p A327623 a:= n-> b(n$3)[2]:
%p A327623 seq(a(n), n=0..23);
%t A327623 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 A327623 a[n_] := b[n, n, n][[2]];
%t A327623 a /@ Range[0, 23] (* _Jean-François Alcover_, Dec 09 2020, after _Alois P. Heinz_ *)
%Y A327623 Main diagonal of A327622.
%Y A327623 Cf. A327619.
%K A327623 nonn
%O A327623 0,4
%A A327623 _Alois P. Heinz_, Sep 19 2019