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

0,9

Links

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

Formula

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

Example

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].

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:= 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 *)

Crossrefs

Column k=8 of A266477.

Sequence in context: A021496 A241830 A151929 * A083236 A345421 A348959

Adjacent sequences: A266688 A266689 A266690 * A266692 A266693 A266694

Keyword

nonn

Author

Emeric Deutsch and Alois P. Heinz, Jan 02 2016

Status

approved