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A327619 Number of parts in all n-times partitions of n.

%I A327619 #17 May 04 2020 12:35:53
%S A327619 0,1,5,25,219,1596,19844,208377,3394835,46799236,886886076,
%T A327619 15668835975,366602236558,7582277939549,199035634246870,
%U A327619 4962275379320665,150339081311823341,4214812414260868163,141823733752997729872,4533014863242019822308,169587948261109794026999
%N A327619 Number of parts in all n-times partitions of n.
%H A327619 Alois P. Heinz, <a href="/A327619/b327619.txt">Table of n, a(n) for n = 0..300</a>
%H A327619 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%e A327619 a(2) = 5: 2, 11, 1|1.
%p A327619 b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
%p A327619      `if`(k=0, [1, 1], `if`(i<2, 0, b(n, i-1, k))+
%p A327619          (h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
%p A327619         b(n-i, min(n-i, i), k)))(b(i$2, k-1))))
%p A327619     end:
%p A327619 a:= n-> b(n$3)[2]:
%p A327619 seq(a(n), n=0..21);
%t A327619 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]]]];
%t A327619 a[n_] := b[n, n, n][[2]];
%t A327619 a /@ Range[0, 21] (* _Jean-François Alcover_, May 01 2020, after Maple *)
%Y A327619 Main diagonal of A327618.
%Y A327619 Cf. A306187, A327623.
%K A327619 nonn
%O A327619 0,3
%A A327619 _Alois P. Heinz_, Sep 19 2019