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A184644 Number of partitions of n having no parts with multiplicity 9.

%I A184644 #14 Jun 12 2025 08:03:59
%S A184644 1,1,2,3,5,7,11,15,22,29,42,55,76,99,133,172,227,290,376,477,612,769,
%T A184644 975,1217,1528,1895,2359,2907,3592,4400,5403,6584,8034,9742,11823,
%U A184644 14272,17234,20713,24897,29803,35674,42542,50719,60272,71592,84794
%N A184644 Number of partitions of n having no parts with multiplicity 9.
%H A184644 Alois P. Heinz, <a href="/A184644/b184644.txt">Table of n, a(n) for n = 0..1000</a>
%F A184644 a(n) = A000041(n) - A183566(n).
%F A184644 a(n) = A183568(n,0) - A183568(n,9).
%F A184644 G.f.: Product_{j>0} (1-x^(9*j)+x^(10*j))/(1-x^j).
%F A184644 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(-9*x) + exp(-11*x)) dx = 0.81120660452677002313966407107688916817839171627473737415672... - _Vaclav Kotesovec_, Jun 12 2025
%p A184644 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
%p A184644       add((l->`if`(j=9, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
%p A184644     end:
%p A184644 a:= n-> (l-> l[1]-l[2])(b(n, n)):
%p A184644 seq(a(n), n=0..50);
%t A184644 b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 9, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
%t A184644 a[n_] := b[n, n][[1]] - b[n, n][[2]];
%t A184644 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)
%Y A184644 Cf. A000041, A183566, A183568, A007690, A116645, A118807, A184639, A184640, A184641, A184642, A184643, A184645.
%K A184644 nonn
%O A184644 0,3
%A A184644 _Alois P. Heinz_, Jan 18 2011