A266687 Number of partitions of n with product of multiplicities of parts equal to 4.

0, 0, 0, 0, 1, 0, 2, 1, 3, 4, 6, 6, 11, 13, 17, 24, 29, 36, 48, 59, 72, 96, 111, 138, 170, 207, 245, 305, 362, 432, 517, 616, 723, 868, 1013, 1194, 1412, 1644, 1915, 2245, 2605, 3019, 3511, 4051, 4677, 5410, 6209, 7125, 8199, 9372, 10718, 12257, 13975, 15902

Offset

0,7

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..18694 (terms 0..10000 from Alois P. Heinz)

Formula

a(n) ~ c * exp(Pi*sqrt(n/3)) * n^(1/4), where c = 0.0108735520090052... - Vaclav Kotesovec, May 24 2018

Example

a(6) = 2: [1,1,1,1,2], [1,1,2,2].
a(7) = 1: [1,1,1,1,3].
a(8) = 3: [2,2,2,2], [1,1,3,3], [1,1,1,1,4].
a(9) = 4: [1,2,2,2,2], [1,1,1,1,2,3], [1,1,2,2,3], [1,1,1,1,5].
a(10) = 6: [1,1,2,3,3], [2,2,3,3], [1,1,1,1,2,4], [1,1,2,2,4], [1,1,4,4], [1,1,1,1,6].

Maple

b:= proc(n, i, p) option remember; `if`(n=0, `if`(p=1, 1, 0),
      `if`(i<1, 0, b(n, i-1, p)+add(`if`(irem(p, j)=0,
       b(n-i*j, i-1, p/j), 0), j=1..n/i)))
    end:
a:= n-> b(n$2, 4):
seq(a(n), n=0..70);

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n == 0, If[p == 1, 1, 0], If[i < 1, 0, b[n, i - 1, p] + Sum[If[Mod[p, j] == 0, b[n - i*j, i - 1, p/j], 0], {j, 1, n/i}]]]; a[n_] := b[n, n, 4]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)

Crossrefs

Column k=4 of A266477.

Sequence in context: A098164 A158504 A293253 * A325271 A338860 A358642

Adjacent sequences: A266684 A266685 A266686 * A266688 A266689 A266690

Keyword

nonn

Author

Emeric Deutsch and Alois P. Heinz, Jan 02 2016

Status

approved