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

Author

Emeric Deutsch and Alois P. Heinz, Jan 02 2016

Keywords

• nonn

Examples

			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].
		

Links

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

Crossrefs

Column k=4 of A266477.

Programs

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 *)

Formula

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