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A268191 The difference between the two largest distinct parts of a partition (0 if no distinct parts), summed over all partitions of n. 2
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified March 26 23:02 EDT 2019. Contains 321565 sequences. (Running on oeis4.)