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

Author

Emeric Deutsch and Alois P. Heinz, Jan 02 2016

Keywords

• nonn

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.

Examples

			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].
		

Links

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

Crossrefs

Column k=5 of A266477.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(i*(p+(i-1)/2)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 *)

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