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A213180 Sum over all partitions lambda of n of Sum_{p:lambda} p^m(p,lambda), where m(p,lambda) is the multiplicity of part p in lambda.

%I A213180 #32 May 24 2018 03:44:00
%S A213180 0,1,3,7,16,28,59,91,170,269,450,655,1162,1602,2527,3793,5805,8034,
%T A213180 12660,17131,26484,37384,53738,73504,114683,153613,221225,313339,
%U A213180 453769,609179,927968,1223909,1804710,2522264,3539835,4855420,7439870,9765555,14009545
%N A213180 Sum over all partitions lambda of n of Sum_{p:lambda} p^m(p,lambda), where m(p,lambda) is the multiplicity of part p in lambda.
%H A213180 Alois P. Heinz, <a href="/A213180/b213180.txt">Table of n, a(n) for n = 0..5000</a>
%F A213180 From _Vaclav Kotesovec_, May 24 2018: (Start)
%F A213180 a(n) ~ c * 3^(n/3), where
%F A213180 c = 5.0144820680945600131204662934686439430547... if mod(n,3)=0
%F A213180 c = 4.6144523178014379613985400559486878971522... if mod(n,3)=1
%F A213180 c = 4.5237761454818383598444208605033385016299... if mod(n,3)=2
%F A213180 (End)
%e A213180 a(6) = 59: (1^6) + (2+1^4) + (2^2+1^2) + (2^3) + (3+1^3) + (3+2+1) + (3^2) + (4+1^2) + (4+2) + (5+1) + (6) = 1+3+5+8+4+6+9+5+6+6+6 = 59.
%p A213180 b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
%p A213180       add((l->`if`(m=0, l, l+[0, l[1]*p^m]))(b(n-p*m, p-1)), m=0..n/p)))
%p A213180     end:
%p A213180 a:= n-> b(n, n)[2]:
%p A213180 seq(a(n), n=0..40);
%t A213180 b[n_, p_] := b[n, p] = If[n==0, {1, 0}, If[p<1, {0, 0}, Sum[Function[l, If[m==0, l, l+{0, l[[1]]*p^m}]][b[n-p*m, p-1]], {m, 0, n/p}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Feb 15 2017, translated from Maple *)
%Y A213180 Cf. A000070 (Sum 1), A006128 (Sum m), A014153 (Sum p), A024786 (Sum floor(1/m)), A066183 (Sum p^2*m), A066186 (Sum p*m), A073336 (Sum floor(m/p)), A116646 (Sum delta(m,2)), A117524 (Sum delta(m,3)), A103628 (Sum delta(m,1)*p), A117525 (Sum delta(m,2)*p), A197126, A213191.
%K A213180 nonn
%O A213180 0,3
%A A213180 _Alois P. Heinz_, Feb 27 2013