A266691 Number of partitions of n with product of multiplicities of parts equal to 8.

0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 3, 2, 5, 4, 10, 9, 17, 20, 25, 31, 47, 53, 71, 89, 109, 138, 171, 205, 257, 317, 375, 461, 557, 664, 792, 962, 1124, 1352, 1596, 1878, 2215, 2621, 3042, 3584, 4180, 4862, 5658, 6593, 7598, 8826, 10190, 11730, 13516, 15562, 17811

Offset: 0

Author

Emeric Deutsch and Alois P. Heinz, Jan 02 2016

Keywords

• nonn

Examples

			a(8) = 2: [1,1,1,1,1,1,1,1], [1,1,1,1,2,2].
a(10) = 3: [1,1,1,1,1,1,1,1,2], [1,1,2,2,2,2], [1,1,1,1,3,3].
a(12) = 5: [1,1,1,1,2,3,3], [1,1,2,2,3,3], [1,1,1,1,1,1,1,1,4], [1,1,1,1,2,2,4], [1,1,1,1,4,4].
		

Links

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

Crossrefs

Column k=8 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:= b(n$2, 8):
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, 8];
Table[a[n], {n, 0, 65}] (* Jean-François Alcover, May 01 2018, translated from Maple *)

Formula

a(n) ~ c * exp(Pi*sqrt(n/3)) * n^(3/4), where c = 0.0012263686774... - Vaclav Kotesovec, May 24 2018