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A116900
Number of partitions of n into at least two parts such that the product of largest and smallest part is equal to n.
6
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, 44, 31, 41, 2, 121, 0, 66, 73, 126, 0, 215, 0, 193, 179, 165, 0, 554, 1, 285, 346, 491, 0, 890, 65, 772, 704, 574, 0, 2330, 0, 847, 1392, 1828, 254, 3212, 0, 2754, 2649, 2282, 0, 7907
OFFSET
0,13
COMMENTS
Clearly a(p)=0, when p is prime.
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
LINKS
FORMULA
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
EXAMPLE
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).
MAPLE
b:= proc(n, i, m) option remember; `if`(n=0, 1,
`if`(i<m, 0, add(b(n-i*j, i-1, m), j=0..n/i)))
end:
a:= n-> add(b(n-d-n/d, d, n/d), d=select(x->
is(x>=sqrt(n)), numtheory[divisors](n))):
seq(a(n), n=0..80); # Alois P. Heinz, Apr 03 2014
MATHEMATICA
f[n_] := Length@ Select[ IntegerPartitions@n, (Length@ # > 1 && Last@# First@# == n) &]; Array[f, 72] (* Robert G. Wilson v, Mar 15 2006 *)
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Giovanni Resta, Mar 14 2006
EXTENSIONS
More terms from Robert G. Wilson v, Mar 15 2006
STATUS
approved