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A266690 Number of partitions of n with product of multiplicities of parts equal to 7. 2
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, May 01 2018, translated from Maple *)

CROSSREFS

Column k=7 of A266477.

Sequence in context: A219029 A090168 A337765 * A230421 A219641 A277758

Adjacent sequences:  A266687 A266688 A266689 * A266691 A266692 A266693

KEYWORD

nonn

AUTHOR

Emeric Deutsch and Alois P. Heinz, Jan 02 2016

STATUS

approved

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Last modified September 30 11:55 EDT 2020. Contains 337439 sequences. (Running on oeis4.)