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
Keywords
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)))).