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A266690 Number of partitions of n with product of multiplicities of parts equal to 7.

%I A266690 #15 May 24 2018 16:10:57
%S A266690 0,0,0,0,0,0,0,1,0,1,1,1,2,2,4,4,5,6,9,10,12,16,20,23,28,33,40,49,59,
%T A266690 69,81,96,112,133,155,181,212,246,284,331,380,438,506,580,666,765,872,
%U A266690 996,1136,1294,1468,1669,1894,2142,2426,2740,3092,3488,3926,4416
%N A266690 Number of partitions of n with product of multiplicities of parts equal to 7.
%C A266690 Also the number of partitions of n such that there is exactly one part which occurs 7 times, while all other parts occur only once.
%H A266690 Alois P. Heinz, <a href="/A266690/b266690.txt">Table of n, a(n) for n = 0..10000</a>
%F A266690 G.f.: Sum_{k>=1} x^(7*k)/(1+x^k) * Product_{j>=1} (1+x^j).
%F A266690 a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(1/4), where c = (60*log(2)-37) / (40*3^(3/4)*Pi) = 0.016019584320... - _Vaclav Kotesovec_, May 24 2018
%e A266690 a(11) = 1: [1,1,1,1,1,1,1,4].
%e A266690 a(12) = 2: [1,1,1,1,1,1,1,2,3], [1,1,1,1,1,1,1,5].
%p A266690 b:= proc(n, i, p) option remember; `if`(i*(p+(i-1)/2)<n, 0, `if`(n=0,
%p A266690       `if`(p=1, 1, 0), b(n, i-1, p) +add(`if`(irem(p, j)>0, 0, (h->
%p A266690        b(h, min(h, i-1), p/j))(n-i*j)), j=1..min(p, n/i))))
%p A266690     end:
%p A266690 a:= n-> b(n$2, 7):
%p A266690 seq(a(n), n=0..65);
%t A266690 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 A266690 a[n_] := b[n, n, 7];
%t A266690 Table[a[n], {n, 0, 65}] (* _Jean-François Alcover_, May 01 2018, translated from Maple *)
%Y A266690 Column k=7 of A266477.
%K A266690 nonn
%O A266690 0,13
%A A266690 _Emeric Deutsch_ and _Alois P. Heinz_, Jan 02 2016