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A184645 Number of partitions of n having no parts with multiplicity 10.

%I A184645 #15 Jun 12 2025 08:05:36
%S A184645 1,1,2,3,5,7,11,15,22,30,41,56,76,100,133,174,227,293,378,482,614,777,
%T A184645 980,1229,1538,1913,2375,2936,3619,4445,5447,6650,8102,9844,11929,
%U A184645 14421,17397,20934,25141,30130,36035,43014,51253,60952,72367,85771,101488
%N A184645 Number of partitions of n having no parts with multiplicity 10.
%C A184645 In general, if k>=1 and g.f. = Product_{j>0} (1 - x^(k*j) + x^((k+1)*j)) / (1-x^j), then 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(-k*x) + exp(-(k+2)*x)) dx. - _Vaclav Kotesovec_, Jun 12 2025
%H A184645 Alois P. Heinz, <a href="/A184645/b184645.txt">Table of n, a(n) for n = 0..1000</a>
%F A184645 a(n) = A000041(n) - A183567(n).
%F A184645 a(n) = A183568(n,0) - A183568(n,10).
%F A184645 G.f.: Product_{j>0} (1-x^(10*j)+x^(11*j))/(1-x^j).
%F A184645 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(-10*x) + exp(-12*x)) dx = 0.81326581550954971049947225462608121545474493920551191360132... - _Vaclav Kotesovec_, Jun 12 2025
%p A184645 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
%p A184645       add((l->`if`(j=10, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
%p A184645     end:
%p A184645 a:= n-> (l-> l[1]-l[2])(b(n, n)):
%p A184645 seq(a(n), n=0..50);
%t A184645 b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 10, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
%t A184645 a[n_] := b[n, n][[1]] - b[n, n][[2]];
%t A184645 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)
%Y A184645 Cf. A000041, A183567, A183568, A007690, A116645, A118807, A184639, A184640, A184641, A184642, A184643, A184644.
%K A184645 nonn
%O A184645 0,3
%A A184645 _Alois P. Heinz_, Jan 18 2011