A266688 Number of partitions of n with product of multiplicities of parts equal to 5.

0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 3, 3, 3, 4, 7, 8, 10, 12, 15, 18, 24, 28, 35, 42, 48, 60, 72, 84, 102, 120, 140, 166, 194, 226, 264, 311, 358, 416, 482, 554, 641, 738, 844, 970, 1112, 1271, 1450, 1654, 1878, 2138, 2429, 2748, 3116, 3524, 3976, 4493, 5065, 5696

Offset

0,11

Comments

Also the number of partitions of n such that there is exactly one part which occurs 5 times, while all other parts occur only once.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Formula

G.f.: Sum_{k>=1} x^(5*k)/(1+x^k) * Product_{j>=1} (1+x^j).

a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(1/4), where c = (12*log(2) - 7) / (8*3^(3/4)*Pi) = 0.023001573808... - Vaclav Kotesovec, May 24 2018

Example

a(9) = 1: [1,1,1,1,1,4].
a(10) = 3: [2,2,2,2,2], [1,1,1,1,1,2,3], [1,1,1,1,1,5].

Maple

b:= proc(n, i, p) option remember; `if`(i*(p+(i-1)/2)<n, 0, `if`(n=0,
      `if`(p=1, 1, 0), b(n, i-1, p) +add(`if`(irem(p, j)>0, 0, (h->
       b(h, min(h, i-1), p/j))(n-i*j)), j=1..min(p, n/i))))
    end:
a:= n-> b(n$2, 5):
seq(a(n), n=0..65);

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[i*(p + (i - 1)/2) < n, 0, If[n == 0, If[p == 1, 1, 0], b[n, i - 1, p] + Sum[If[Mod[p, j] > 0, 0, Function[h, b[h, Min[h, i - 1], p/j]][n - i*j]], {j, 1, Min[p, n/i]}]]];
a[n_] := b[n, n, 5];
Table[a[n], {n, 0, 65}] (* Jean-François Alcover, May 01 2018, translated from Maple *)

Crossrefs

Column k=5 of A266477.

Sequence in context: A035567 A219328 A108688 * A347139 A285286 A123371

Adjacent sequences: A266685 A266686 A266687 * A266689 A266690 A266691

Keyword

nonn

Author

Emeric Deutsch and Alois P. Heinz, Jan 02 2016

Status

approved