A266690 Number of partitions of n with product of multiplicities of parts equal to 7.

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 2, 2, 4, 4, 5, 6, 9, 10, 12, 16, 20, 23, 28, 33, 40, 49, 59, 69, 81, 96, 112, 133, 155, 181, 212, 246, 284, 331, 380, 438, 506, 580, 666, 765, 872, 996, 1136, 1294, 1468, 1669, 1894, 2142, 2426, 2740, 3092, 3488, 3926, 4416

Offset

0,13

Comments

Also the number of partitions of n such that there is exactly one part which occurs 7 times, while all other parts occur only once.

Links

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

Formula

G.f.: Sum_{k>=1} x^(7*k)/(1+x^k) * Product_{j>=1} (1+x^j).

a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(1/4), where c = (60*log(2)-37) / (40*3^(3/4)*Pi) = 0.016019584320... - Vaclav Kotesovec, May 24 2018

Example

a(11) = 1: [1,1,1,1,1,1,1,4].
a(12) = 2: [1,1,1,1,1,1,1,2,3], [1,1,1,1,1,1,1,5].

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, 7):
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, 7];
Table[a[n], {n, 0, 65}] (* Jean-François Alcover, May 01 2018, translated from Maple *)

Crossrefs

Column k=7 of A266477.

Sequence in context: A342498 A340278 A337765 * A230421 A219641 A341464

Adjacent sequences: A266687 A266688 A266689 * A266691 A266692 A266693

Keyword

nonn

Author

Emeric Deutsch and Alois P. Heinz, Jan 02 2016

Status

approved