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A184640 Number of partitions of n having no parts with multiplicity 5.

%I A184640 #13 Jun 12 2025 08:01:29
%S A184640 1,1,2,3,5,6,11,14,21,28,39,51,72,92,124,160,210,266,349,438,562,704,
%T A184640 892,1107,1395,1720,2141,2631,3249,3965,4873,5916,7216,8730,10585,
%U A184640 12742,15387,18443,22151,26466,31646,37659,44873,53212,63149,74666,88295
%N A184640 Number of partitions of n having no parts with multiplicity 5.
%H A184640 Alois P. Heinz, <a href="/A184640/b184640.txt">Table of n, a(n) for n = 0..1000</a>
%F A184640 a(n) = A000041(n) - A183562(n).
%F A184640 a(n) = A183568(n,0) - A183568(n,5).
%F A184640 G.f.: Product_{j>0} (1-x^(5*j)+x^(6*j))/(1-x^j).
%F A184640 a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-5*x) + exp(-7*x)) dx = 0.78834765570757713777493985857868631321765157344539753651545... - _Vaclav Kotesovec_, Jun 12 2025
%e A184640 a(5) = 6, because 6 partitions of 5 have no parts with multiplicity 5: [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5].
%p A184640 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
%p A184640       add((l->`if`(j=5, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
%p A184640     end:
%p A184640 a:= n-> (l-> l[1]-l[2])(b(n, n)):
%p A184640 seq(a(n), n=0..50);
%t A184640 b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 5, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
%t A184640 a[n_] := b[n, n][[1]] - b[n, n][[2]];
%t A184640 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)
%Y A184640 Cf. A000041, A183562, A183568, A007690, A116645, A118807, A184639, A184641, A184642, A184643, A184644, A184645.
%K A184640 nonn
%O A184640 0,3
%A A184640 _Alois P. Heinz_, Jan 18 2011