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Number of partitions of n into at least two parts such that the product of largest and smallest part is equal to n.
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%I #23 Dec 28 2015 05:36:20

%S 0,0,0,0,1,0,0,0,1,1,1,0,2,0,2,1,5,0,7,0,7,4,8,0,21,1,14,13,25,0,43,0,

%T 44,31,41,2,121,0,66,73,126,0,215,0,193,179,165,0,554,1,285,346,491,0,

%U 890,65,772,704,574,0,2330,0,847,1392,1828,254,3212,0,2754,2649,2282,0,7907

%N Number of partitions of n into at least two parts such that the product of largest and smallest part is equal to n.

%C Clearly a(p)=0, when p is prime.

%C Number of partitions p of n such that mean(p) = multiplicity(max(p)). For example, a(12) counts these two partitions: 441111, 332211. See the Mathematica program at A240200 for a count of partitions defined in this manner, along with related sequences. - _Clark Kimberling_, Apr 03 2014

%H Alois P. Heinz, <a href="/A116900/b116900.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = coefficient of x^n in expansion of Sum_{d|n} x^(d+n/d)/Product(1-x^k, k=d..n/d). - _Vladeta Jovovic_, Nov 24 2008

%e a(21) = 4 since property holds for 4 partitions of 21: (7,7,4,3), (7,6,5,3), (7,5,3,3,3), (7,4,4,3,3).

%p b:= proc(n, i, m) option remember; `if`(n=0, 1,

%p `if`(i<m, 0, add(b(n-i*j, i-1, m), j=0..n/i)))

%p end:

%p a:= n-> add(b(n-d-n/d, d, n/d), d=select(x->

%p is(x>=sqrt(n)), numtheory[divisors](n))):

%p seq(a(n), n=0..80); # _Alois P. Heinz_, Apr 03 2014

%t f[n_] := Length@ Select[ IntegerPartitions@n, (Length@ # > 1 && Last@# First@# == n) &]; Array[f, 72] (* _Robert G. Wilson v_, Mar 15 2006 *)

%t b[n_, i_, m_] := b[n, i, m] = If[n==0, 1, If[i<m, 0, Sum[b[n-i*j, i-1, m], {j, 0, n/i}]]]; a[n_] := Sum[b[n-d-n/d, d, n/d], {d, Select[Divisors[n], # >= Sqrt[n]&]}]; a[0] = 0; Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Sep 12 2015, after _Alois P. Heinz_ *)

%Y Cf. A000041, A116901, A116902, A240200.

%K nonn

%O 0,13

%A _Giovanni Resta_, Mar 14 2006

%E More terms from _Robert G. Wilson v_, Mar 15 2006