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A336516 Sum of parts, counted without multiplicity, in all compositions of n.

%I A336516 #26 Mar 11 2022 09:55:23
%S A336516 0,1,3,10,24,59,136,309,682,1493,3223,6904,14675,31013,65202,136512,
%T A336516 284748,592082,1227709,2539516,5241640,10798133,22206568,45597489,
%U A336516 93495667,191464970,391636718,800233551,1633530732,3331568080,6789078236,13824212219,28129459098
%N A336516 Sum of parts, counted without multiplicity, in all compositions of n.
%H A336516 Alois P. Heinz, <a href="/A336516/b336516.txt">Table of n, a(n) for n = 0..1000</a>
%e A336516 a(4) = 1 + 1 + 2 + 1 + 2 + 1 + 2 + 2 + 1 + 3 + 3 + 1 + 4 = 24: (1)111, (1)1(2), (1)(2)1, (2)(1)1, (2)2, (1)(3), (3)(1), (4).
%p A336516 b:= proc(n, i, p) option remember; `if`(n=0, [p!, 0],
%p A336516       `if`(i<1, 0, add((p-> [0, `if`(j=0, 0, p[1]*i)]+p)(
%p A336516          b(n-i*j, i-1, p+j)/j!), j=0..n/i)))
%p A336516     end:
%p A336516 a:= n-> b(n$2, 0)[2]:
%p A336516 seq(a(n), n=0..38);
%t A336516 b[n_, i_, p_] := b[n, i, p] = If[n == 0, {p!, 0},
%t A336516      If[i < 1, {0, 0}, Sum[Function[{0, If[j == 0, 0, #[[1]]*i]} + #][
%t A336516        b[n - i*j, i - 1, p + j]/j!], {j, 0, n/i}]]];
%t A336516 a[n_] := b[n, n, 0][[2]];
%t A336516 Table[a[n], {n, 0, 38}] (* _Jean-François Alcover_, Mar 11 2022, after _Alois P. Heinz_ *)
%Y A336516 Cf. A001787 (all parts), A014153 (the same for partitions), A336511, A336512, A336579, A336875.
%K A336516 nonn
%O A336516 0,3
%A A336516 _Alois P. Heinz_, Jul 24 2020