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A266692 Number of partitions of n with product of multiplicities of parts equal to 9.

%I A266692 #11 May 24 2018 09:50:48
%S A266692 0,0,0,0,0,0,0,0,0,2,0,1,3,2,4,5,7,7,14,13,18,25,29,35,47,54,65,86,
%T A266692 101,120,147,174,205,254,291,347,419,486,565,676,779,908,1065,1228,
%U A266692 1425,1668,1906,2198,2547,2912,3336,3841,4384,4998,5728,6513,7404,8436
%N A266692 Number of partitions of n with product of multiplicities of parts equal to 9.
%H A266692 Alois P. Heinz, <a href="/A266692/b266692.txt">Table of n, a(n) for n = 0..10000</a>
%F A266692 a(n) ~ c * exp(Pi*sqrt(n/3)) * n^(1/4), where c = 0.004308121528... - _Vaclav Kotesovec_, May 24 2018
%e A266692 a(9) = 2: [1,1,1,1,1,1,1,1,1], [1,1,1,2,2,2].
%e A266692 a(11) = 1: [1,1,1,1,1,1,1,1,1,2].
%e A266692 a(12) = 3: [1,1,1,1,1,1,1,1,1,3], [1,1,1,2,2,2,3], [1,1,1,3,3,3].
%p A266692 b:= proc(n, i, p) option remember; `if`(i*(p+(i-1)/2)<n, 0, `if`(n=0,
%p A266692       `if`(p=1, 1, 0), b(n, i-1, p) +add(`if`(irem(p, j)>0, 0, (h->
%p A266692        b(h, min(h, i-1), p/j))(n-i*j)), j=1..min(p, n/i))))
%p A266692     end:
%p A266692 a:= b(n$2, 9):
%p A266692 seq(a(n), n=0..65);
%t A266692 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 A266692 a[n_] := b[n, n, 9];
%t A266692 Table[a[n], {n, 0, 65}] (* _Jean-François Alcover_, May 01 2018, translated from Maple *)
%Y A266692 Column k=9 of A266477.
%K A266692 nonn
%O A266692 0,10
%A A266692 _Emeric Deutsch_ and _Alois P. Heinz_, Jan 02 2016