OFFSET
0,6
COMMENTS
Number of partitions of n such that the least part is equal to the difference between the 2 largest distinct parts. These are the conjugates of the partitions in the definition. Example: a(6) = 4 because we have [1,1,1,1,2], [1,1,2,2], [1,2,3], and [2,4]. - Emeric Deutsch , Feb 26 2016
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: Sum_{k>=1} Sum_{j>=2} Sum_{i>=1} x^{k*(i+j)}/Product_{p=i+1..j-1} (1-x^p).
EXAMPLE
a(2) = 0 because neither [2] nor [1,1] satisfy the required conditions.
a(6) = 4 because we have [1,5], [2,4], [1,2,3], and [1,1,2,2].
MAPLE
g := sum(sum(sum(x^(k*(i+j))/(product(1-x^p, p = i+1 .. j-1)), i = 1 .. j-1), j = 2 .. 50), k = 1 .. 50): gser := series(g, x = 0, 70): seq(coeff(gser, x, m), m = 0 .. 65);
MATHEMATICA
okQ[p_] := With[{m = Min[p], M = Max[p]}, If[m == M, False, Count[p, m] == Count[p, M]]]; a[0] = 0; a[n_] := Select[IntegerPartitions[n], okQ] // Length; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 50}] (* Jean-François Alcover, Jan 24 2016 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jan 24 2016
STATUS
approved