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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
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
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: A383094 A340278 A337765 * A230421 A219641 A341464
KEYWORD
nonn
AUTHOR
STATUS
approved