A268191 The difference between the two largest distinct parts of a partition (0 if no distinct parts), summed over all partitions of n.

0, 0, 1, 3, 8, 14, 27, 42, 67, 101, 149, 210, 301, 416, 565, 770, 1030, 1368, 1800, 2357, 3055, 3962, 5068, 6485, 8232, 10444, 13125, 16506, 20600, 25701, 31865, 39483, 48644, 59906, 73375, 89846, 109515, 133379, 161784, 196078, 236801, 285720, 343623, 412866, 494624, 591991, 706623, 842625

Offset: 1

Author

Emeric Deutsch, Feb 10 2016

Keywords

• nonn

Examples

			a(5) = 8 because the partitions [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], and [1,1,1,1,1] of 5 contribute 0, 3, 1, 2, 1, 1, and 0, respectively.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..1932 (terms 1..1000 from Alois P. Heinz)

Crossrefs

Cf. A268190.

Programs

Maple

g := add(add((i-j)*x^(i+j)/((1-x^i)*mul(1-x^k, k = 1 .. j)), j = 1 .. i-1), i = 2 .. 80): gser := series(g, x = 0, 55): seq(coeff(gser, x, n), n = 1 .. 50);
# second Maple program:
b:= proc(n, l, i) option remember; `if`(irem(n, i)=0,
      `if`(l=0, 0, i-l), 0) +`if`(i>n, 0, add(b(n-i*j,
      `if`(j=0, l, i), i+1), j=0..(n-1)/i))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 11 2016

Mathematica

b[n_, l_, i_] := b[n, l, i] = If[Mod[n, i] == 0, If[l == 0, 0, i-l], 0] + If[i>n, 0, Sum[b[n-i*j, If[j == 0, l, i], i+1], {j, 0, (n-1)/i}]]; a[n_] := b[n, 0, 1]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)

Formula

a(n) = Sum_{k>0} k*A268190(n,k).

G.f.: g = Sum_{i>1} (Sum_{j = 1..i-1} ((i-j)*x^{i+j}/((1-x^i) * Product_{k=1..j} (1 - x^k)))).