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A266687 Number of partitions of n with product of multiplicities of parts equal to 4.

%I A266687 #20 May 25 2018 07:00:51
%S A266687 0,0,0,0,1,0,2,1,3,4,6,6,11,13,17,24,29,36,48,59,72,96,111,138,170,
%T A266687 207,245,305,362,432,517,616,723,868,1013,1194,1412,1644,1915,2245,
%U A266687 2605,3019,3511,4051,4677,5410,6209,7125,8199,9372,10718,12257,13975,15902
%N A266687 Number of partitions of n with product of multiplicities of parts equal to 4.
%H A266687 Vaclav Kotesovec, <a href="/A266687/b266687.txt">Table of n, a(n) for n = 0..18694</a> (terms 0..10000 from Alois P. Heinz)
%F A266687 a(n) ~ c * exp(Pi*sqrt(n/3)) * n^(1/4), where c = 0.0108735520090052... - _Vaclav Kotesovec_, May 24 2018
%e A266687 a(6) = 2: [1,1,1,1,2], [1,1,2,2].
%e A266687 a(7) = 1: [1,1,1,1,3].
%e A266687 a(8) = 3: [2,2,2,2], [1,1,3,3], [1,1,1,1,4].
%e A266687 a(9) = 4: [1,2,2,2,2], [1,1,1,1,2,3], [1,1,2,2,3], [1,1,1,1,5].
%e A266687 a(10) = 6: [1,1,2,3,3], [2,2,3,3], [1,1,1,1,2,4], [1,1,2,2,4], [1,1,4,4], [1,1,1,1,6].
%p A266687 b:= proc(n, i, p) option remember; `if`(n=0, `if`(p=1, 1, 0),
%p A266687       `if`(i<1, 0, b(n, i-1, p)+add(`if`(irem(p, j)=0,
%p A266687        b(n-i*j, i-1, p/j), 0), j=1..n/i)))
%p A266687     end:
%p A266687 a:= n-> b(n$2, 4):
%p A266687 seq(a(n), n=0..70);
%t A266687 b[n_, i_, p_] := b[n, i, p] = If[n == 0, If[p == 1, 1, 0], If[i < 1, 0, b[n, i - 1, p] + Sum[If[Mod[p, j] == 0, b[n - i*j, i - 1, p/j], 0], {j, 1, n/i}]]]; a[n_] := b[n, n, 4]; Table[a[n], {n, 0, 70}] (* _Jean-François Alcover_, Dec 22 2016, after _Alois P. Heinz_ *)
%Y A266687 Column k=4 of A266477.
%K A266687 nonn
%O A266687 0,7
%A A266687 _Emeric Deutsch_ and _Alois P. Heinz_, Jan 02 2016