A266687 - OEIS

%I #20 May 25 2018 07:00:51

%S 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 207,245,305,362,432,517,616,723,868,1013,1194,1412,1644,1915,2245,

%U 2605,3019,3511,4051,4677,5410,6209,7125,8199,9372,10718,12257,13975,15902

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

%H 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 a(n) ~ c * exp(Pi*sqrt(n/3)) * n^(1/4), where c = 0.0108735520090052... - _Vaclav Kotesovec_, May 24 2018

%e a(6) = 2: [1,1,1,1,2], [1,1,2,2].

%e a(7) = 1: [1,1,1,1,3].

%e a(8) = 3: [2,2,2,2], [1,1,3,3], [1,1,1,1,4].

%e 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 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 b:= proc(n, i, p) option remember; `if`(n=0, `if`(p=1, 1, 0),

%p       `if`(i<1, 0, b(n, i-1, p)+add(`if`(irem(p, j)=0,

%p        b(n-i*j, i-1, p/j), 0), j=1..n/i)))

%p     end:

%p a:= n-> b(n$2, 4):

%p seq(a(n), n=0..70);

%t 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 Column k=4 of A266477.

%K nonn

%O 0,7

%A _Emeric Deutsch_ and _Alois P. Heinz_, Jan 02 2016