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

0, 0, 3, 11, 36, 79, 186, 345, 672, 1163, 2026, 3273, 5388, 8301, 12912, 19349, 28961, 42071, 61253, 86921, 123404, 171972, 239020, 327386, 447743, 604255, 813645, 1084657, 1441643, 1899450, 2496510, 3255653, 4234822, 5472953, 7053217, 9038784, 11554020

Offset: 0

Author

Alois P. Heinz, Mar 04 2013

Keywords

• nonn

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Column k=0 of A222730.

Cf. A000041, A000217, A014153.

Programs

Maple

b:= proc(n, p) option remember; `if`(n=0 and p=0, [1, 0], `if`(p<1, [0$2],
      add((l->`if`(m=0, l+[0, l[1]*p], l))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..55);

Mathematica

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 *)

Formula

a(n) = A000217(n)*A000041(n)-A014153(n-1).