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A184643 Number of partitions of n having no parts with multiplicity 8.

%I A184643 #13 Jun 12 2025 08:03:25
%S A184643 1,1,2,3,5,7,11,15,21,30,41,55,75,99,131,172,223,288,372,474,603,764,
%T A184643 962,1206,1509,1876,2326,2878,3543,4351,5330,6506,7921,9623,11655,
%U A184643 14085,16987,20434,24529,29392,35138,41930,49947,59381,70474,83512,98779
%N A184643 Number of partitions of n having no parts with multiplicity 8.
%H A184643 Alois P. Heinz, <a href="/A184643/b184643.txt">Table of n, a(n) for n = 0..1000</a>
%F A184643 a(n) = A000041(n) - A183565(n).
%F A184643 a(n) = A183568(n,0) - A183568(n,8).
%F A184643 G.f.: Product_{j>0} (1-x^(8*j)+x^(9*j))/(1-x^j).
%F A184643 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(-8*x) + exp(-10*x)) dx = 0.80836901097063952622501649557292291036896118821761722817375... - _Vaclav Kotesovec_, Jun 12 2025
%p A184643 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
%p A184643       add((l->`if`(j=8, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
%p A184643     end:
%p A184643 a:= n-> (l-> l[1]-l[2])(b(n, n)):
%p A184643 seq(a(n), n=0..50);
%t A184643 b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 8, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
%t A184643 a[n_] := b[n, n][[1]] - b[n, n][[2]];
%t A184643 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)
%Y A184643 Cf. A000041, A183565, A183568, A007690, A116645, A118807, A184639, A184640, A184641, A184642, A184644, A184645.
%K A184643 nonn
%O A184643 0,3
%A A184643 _Alois P. Heinz_, Jan 18 2011