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A266689 Number of partitions of n with product of multiplicities of parts equal to 6.

%I A266689 #12 May 24 2018 09:49:16
%S A266689 0,0,0,0,0,0,1,1,2,2,2,7,6,12,11,16,22,32,35,51,61,70,95,118,144,177,
%T A266689 222,257,313,382,459,547,664,770,933,1092,1275,1513,1786,2070,2431,
%U A266689 2838,3287,3830,4435,5094,5918,6825,7821,9010,10340,11820,13525,15474
%N A266689 Number of partitions of n with product of multiplicities of parts equal to 6.
%H A266689 Alois P. Heinz, <a href="/A266689/b266689.txt">Table of n, a(n) for n = 0..10000</a>
%F A266689 a(n) ~ c * exp(Pi*sqrt(n/3)) * n^(1/4), where c = 0.01368862060... - _Vaclav Kotesovec_, May 24 2018
%e A266689 a(7) = 1: [1,1,1,2,2].
%e A266689 a(8) = 2: [1,1,1,1,1,1,2], [1,1,2,2,2].
%e A266689 a(11) = 7: [1,1,1,1,1,1,2,3], [1,1,2,2,2,3], [1,1,1,2,3,3], [1,1,3,3,3], [1,1,1,2,2,4], [1,1,1,4,4], [1,1,1,1,1,1,5].
%p A266689 b:= proc(n, i, p) option remember; `if`(i*(p+(i-1)/2)<n, 0, `if`(n=0,
%p A266689       `if`(p=1, 1, 0), b(n, i-1, p) +add(`if`(irem(p, j)>0, 0, (h->
%p A266689        b(h, min(h, i-1), p/j))(n-i*j)), j=1..min(p, n/i))))
%p A266689     end:
%p A266689 a:= n-> b(n$2, 6):
%p A266689 seq(a(n), n=0..65);
%t A266689 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 A266689 a[n_] := b[n, n, 6];
%t A266689 Table[a[n], {n, 0, 65}] (* _Jean-François Alcover_, May 01 2018, translated from Maple *)
%Y A266689 Column k=6 of A266477.
%K A266689 nonn
%O A266689 0,9
%A A266689 _Emeric Deutsch_ and _Alois P. Heinz_, Jan 02 2016