A266689 Number of partitions of n with product of multiplicities of parts equal to 6.

0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 2, 7, 6, 12, 11, 16, 22, 32, 35, 51, 61, 70, 95, 118, 144, 177, 222, 257, 313, 382, 459, 547, 664, 770, 933, 1092, 1275, 1513, 1786, 2070, 2431, 2838, 3287, 3830, 4435, 5094, 5918, 6825, 7821, 9010, 10340, 11820, 13525, 15474

Offset

0,9

Links

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

Formula

a(n) ~ c * exp(Pi*sqrt(n/3)) * n^(1/4), where c = 0.01368862060... - Vaclav Kotesovec, May 24 2018

Example

a(7) = 1: [1,1,1,2,2].
a(8) = 2: [1,1,1,1,1,1,2], [1,1,2,2,2].
a(11) = 7: [1,1,1,1,1,1,2,3], [1,1,2,2,2,3], [1,1,1,2,3,3], [1,1,3,3,3], [1,1,1,2,2,4], [1,1,1,4,4], [1,1,1,1,1,1,5].

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, 6):
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, 6];
Table[a[n], {n, 0, 65}] (* Jean-François Alcover, May 01 2018, translated from Maple *)

Crossrefs

Column k=6 of A266477.

Sequence in context: A094246 A169592 A245600 * A265988 A340976 A023573

Adjacent sequences: A266686 A266687 A266688 * A266690 A266691 A266692

Keyword

nonn

Author

Emeric Deutsch and Alois P. Heinz, Jan 02 2016

Status

approved