A266693 Number of partitions of n with product of multiplicities of parts equal to 10.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 3, 4, 2, 7, 8, 12, 13, 19, 22, 32, 36, 46, 64, 72, 88, 112, 134, 160, 203, 236, 287, 343, 412, 477, 577, 676, 798, 944, 1101, 1283, 1516, 1754, 2030, 2361, 2738, 3157, 3657, 4202, 4826, 5567, 6356, 7279, 8340, 9494, 10815

Offset

0,13

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 = 0.007782666499... - Vaclav Kotesovec, May 24 2018

Example

a(9) = 1: [1,1,1,1,1,2,2].
a(12) = 3: [1,1,1,1,1,1,1,1,1,1,2], [1,1,2,2,2,2,2], [1,1,1,1,1,2,2,3].
a(13) = 4: [1,1,1,1,1,1,1,1,1,1,3], [1,1,1,1,1,2,3,3], [1,1,1,1,1,2,2,4], [1,1,1,1,1,4,4].

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, 10):
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, 10];
Table[a[n], {n, 0, 65}] (* Jean-François Alcover, May 01 2018, translated from Maple *)

Crossrefs

Column k=10 of A266477.

Sequence in context: A045901 A098003 A026245 * A026178 A026197 A026217

Adjacent sequences: A266690 A266691 A266692 * A266694 A266695 A266696

Keyword

nonn

Author

Emeric Deutsch and Alois P. Heinz, Jan 02 2016

Status

approved