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A266691 Number of partitions of n with product of multiplicities of parts equal to 8.

%I A266691 #11 May 24 2018 09:50:20
%S A266691 0,0,0,0,0,0,0,0,2,0,3,2,5,4,10,9,17,20,25,31,47,53,71,89,109,138,171,
%T A266691 205,257,317,375,461,557,664,792,962,1124,1352,1596,1878,2215,2621,
%U A266691 3042,3584,4180,4862,5658,6593,7598,8826,10190,11730,13516,15562,17811
%N A266691 Number of partitions of n with product of multiplicities of parts equal to 8.
%H A266691 Alois P. Heinz, <a href="/A266691/b266691.txt">Table of n, a(n) for n = 0..10000</a>
%F A266691 a(n) ~ c * exp(Pi*sqrt(n/3)) * n^(3/4), where c = 0.0012263686774... - _Vaclav Kotesovec_, May 24 2018
%e A266691 a(8) = 2: [1,1,1,1,1,1,1,1], [1,1,1,1,2,2].
%e A266691 a(10) = 3: [1,1,1,1,1,1,1,1,2], [1,1,2,2,2,2], [1,1,1,1,3,3].
%e A266691 a(12) = 5: [1,1,1,1,2,3,3], [1,1,2,2,3,3], [1,1,1,1,1,1,1,1,4], [1,1,1,1,2,2,4], [1,1,1,1,4,4].
%p A266691 b:= proc(n, i, p) option remember; `if`(i*(p+(i-1)/2)<n, 0, `if`(n=0,
%p A266691       `if`(p=1, 1, 0), b(n, i-1, p) +add(`if`(irem(p, j)>0, 0, (h->
%p A266691        b(h, min(h, i-1), p/j))(n-i*j)), j=1..min(p, n/i))))
%p A266691     end:
%p A266691 a:= b(n$2, 8):
%p A266691 seq(a(n), n=0..65);
%t A266691 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 A266691 a[n_] := b[n, n, 8];
%t A266691 Table[a[n], {n, 0, 65}] (* _Jean-François Alcover_, May 01 2018, translated from Maple *)
%Y A266691 Column k=8 of A266477.
%K A266691 nonn
%O A266691 0,9
%A A266691 _Emeric Deutsch_ and _Alois P. Heinz_, Jan 02 2016