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A184641 Number of partitions of n having no parts with multiplicity 6.

%I A184641 #13 Jun 12 2025 08:02:11
%S A184641 1,1,2,3,5,7,10,15,21,29,40,54,72,96,127,166,216,279,358,457,580,735,
%T A184641 924,1159,1446,1799,2228,2752,3388,4158,5087,6207,7551,9165,11093,
%U A184641 13401,16144,19412,23286,27882,33310,39727,47289,56191,66647,78923,93299
%N A184641 Number of partitions of n having no parts with multiplicity 6.
%H A184641 Alois P. Heinz, <a href="/A184641/b184641.txt">Table of n, a(n) for n = 0..1000</a>
%F A184641 a(n) = A000041(n) - A183563(n).
%F A184641 a(n) = A183568(n,0) - A183568(n,6).
%F A184641 G.f.: Product_{j>0} (1-x^(6*j)+x^(7*j))/(1-x^j).
%F A184641 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(-6*x) + exp(-8*x)) dx = 0.79818518024793359047735154473665146019665210453617381247423... - _Vaclav Kotesovec_, Jun 12 2025
%e A184641 a(6) = 10, because 10 partitions of 6 have no parts with multiplicity 6: [1,1,1,1,2], [1,1,2,2], [2,2,2], [1,1,1,3], [1,2,3], [3,3], [1,1,4], [2,4], [1,5], [6].
%p A184641 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
%p A184641       add((l->`if`(j=6, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
%p A184641     end:
%p A184641 a:= n-> (l-> l[1]-l[2])(b(n, n)):
%p A184641 seq(a(n), n=0..50);
%t A184641 b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 6, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
%t A184641 a[n_] := b[n, n][[1]] - b[n, n][[2]];
%t A184641 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)
%Y A184641 Cf. A000041, A183563, A183568, A007690, A116645, A118807, A184639, A184640, A184642, A184643, A184644, A184645.
%K A184641 nonn
%O A184641 0,3
%A A184641 _Alois P. Heinz_, Jan 18 2011