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%I #15 May 24 2018 16:07:23
%S 0,0,0,0,0,1,0,1,1,1,3,3,3,4,7,8,10,12,15,18,24,28,35,42,48,60,72,84,
%T 102,120,140,166,194,226,264,311,358,416,482,554,641,738,844,970,1112,
%U 1271,1450,1654,1878,2138,2429,2748,3116,3524,3976,4493,5065,5696
%N Number of partitions of n with product of multiplicities of parts equal to 5.
%C Also the number of partitions of n such that there is exactly one part which occurs 5 times, while all other parts occur only once.
%H Alois P. Heinz, <a href="/A266688/b266688.txt">Table of n, a(n) for n = 0..10000</a>
%F G.f.: Sum_{k>=1} x^(5*k)/(1+x^k) * Product_{j>=1} (1+x^j).
%F a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(1/4), where c = (12*log(2) - 7) / (8*3^(3/4)*Pi) = 0.023001573808... - _Vaclav Kotesovec_, May 24 2018
%e a(9) = 1: [1,1,1,1,1,4].
%e a(10) = 3: [2,2,2,2,2], [1,1,1,1,1,2,3], [1,1,1,1,1,5].
%p b:= proc(n, i, p) option remember; `if`(i*(p+(i-1)/2)<n, 0, `if`(n=0,
%p `if`(p=1, 1, 0), b(n, i-1, p) +add(`if`(irem(p, j)>0, 0, (h->
%p b(h, min(h, i-1), p/j))(n-i*j)), j=1..min(p, n/i))))
%p end:
%p a:= n-> b(n$2, 5):
%p seq(a(n), n=0..65);
%t 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]}]]];
%t a[n_] := b[n, n, 5];
%t Table[a[n], {n, 0, 65}] (* _Jean-François Alcover_, May 01 2018, translated from Maple *)
%Y Column k=5 of A266477.
%K nonn
%O 0,11
%A _Emeric Deutsch_ and _Alois P. Heinz_, Jan 02 2016
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