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

1,4

Links

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

Example

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.

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 *)

Crossrefs

Cf. A268190.

Sequence in context: A295200 A321047 A123329 * A169929 A129067 A298612

Adjacent sequences: A268188 A268189 A268190 * A268192 A268193 A268194

Keyword

nonn

Author

Emeric Deutsch, Feb 10 2016

Status

approved