|
|
A265259
|
|
Number of partitions of n such that least and largest parts are distinct and occur the same number of times.
|
|
1
|
|
|
0, 0, 0, 1, 1, 2, 4, 4, 7, 9, 14, 14, 26, 25, 40, 47, 65, 71, 108, 114, 163, 188, 246, 280, 383, 429, 555, 653, 825, 950, 1215, 1393, 1741, 2036, 2488, 2906, 3571, 4128, 5003, 5853, 7032, 8169, 9805, 11370, 13544, 15765, 18621
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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
|
|
|
STATUS
|
approved
|
|
|
|