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A213679 Total sum of parts <= n of multiplicity 0 in all partitions of n.

%I A213679 #13 Jan 24 2014 09:49:20
%S A213679 0,0,3,11,36,79,186,345,672,1163,2026,3273,5388,8301,12912,19349,
%T A213679 28961,42071,61253,86921,123404,171972,239020,327386,447743,604255,
%U A213679 813645,1084657,1441643,1899450,2496510,3255653,4234822,5472953,7053217,9038784,11554020
%N A213679 Total sum of parts <= n of multiplicity 0 in all partitions of n.
%H A213679 Alois P. Heinz, <a href="/A213679/b213679.txt">Table of n, a(n) for n = 0..1000</a>
%F A213679 a(n) = A000217(n)*A000041(n)-A014153(n-1).
%e A213679 The partitions of n=4 are [1,1,1,1], [2,1,1], [2,2], [3,1], [4].  Parts <= 4 with multiplicity m=0 sum up to (2+3+4)+(3+4)+(1+3+4)+(2+4)+(1+2+3) = 36, thus a(4) = 36.
%p A213679 b:= proc(n, p) option remember; `if`(n=0 and p=0, [1, 0], `if`(p<1, [0$2],
%p A213679       add((l->`if`(m=0, l+[0, l[1]*p], l))(b(n-p*m, p-1)), m=0..n/p)))
%p A213679     end:
%p A213679 a:= n-> b(n, n)[2]:
%p A213679 seq(a(n), n=0..55);
%t A213679 b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 55}] (* _Jean-François Alcover_, Jan 24 2014, after _Alois P. Heinz_ *)
%Y A213679 Column k=0 of A222730.
%Y A213679 Cf. A000041, A000217, A014153.
%K A213679 nonn
%O A213679 0,3
%A A213679 _Alois P. Heinz_, Mar 04 2013