A265259 Number of partitions of n such that least and largest parts are distinct and occur the same number of times.

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

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

Sequence in context: A241387 A284612 A070072 * A238493 A362937 A268781

Adjacent sequences: A265256 A265257 A265258 * A265260 A265261 A265262

Keyword

nonn

Author

Emeric Deutsch, Jan 24 2016

Status

approved