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Number of partitions of n such that least and largest parts are distinct and occur the same number of times.
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%I #17 Feb 27 2016 03:46:17

%S 0,0,0,1,1,2,4,4,7,9,14,14,26,25,40,47,65,71,108,114,163,188,246,280,

%T 383,429,555,653,825,950,1215,1393,1741,2036,2488,2906,3571,4128,5003,

%U 5853,7032,8169,9805,11370,13544,15765,18621

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

%C 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

%H Alois P. Heinz, <a href="/A265259/b265259.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: Sum_{k>=1} Sum_{j>=2} Sum_{i>=1} x^{k*(i+j)}/Product_{p=i+1..j-1} (1-x^p).

%e a(2) = 0 because neither [2] nor [1,1] satisfy the required conditions.

%e a(6) = 4 because we have [1,5], [2,4], [1,2,3], and [1,1,2,2].

%p 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);

%t 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 *)

%K nonn

%O 0,6

%A _Emeric Deutsch_, Jan 24 2016