cp's OEIS Frontend

A184639 Number of partitions of n having no parts with multiplicity 4.

%I A184639 #14 Jun 12 2025 08:00:50
%S A184639 1,1,2,3,4,7,10,14,19,27,37,50,67,88,115,153,196,253,324,412,524,661,
%T A184639 828,1036,1290,1603,1980,2443,2997,3671,4487,5460,6631,8034,9703,
%U A184639 11703,14075,16890,20226,24175,28838,34332,40801,48394,57307,67765,79974
%N A184639 Number of partitions of n having no parts with multiplicity 4.
%H A184639 Alois P. Heinz, <a href="/A184639/b184639.txt">Table of n, a(n) for n = 0..1000</a>
%F A184639 a(n) = A000041(n) - A183561(n).
%F A184639 a(n) = A183568(n,0) - A183568(n,4).
%F A184639 G.f.: Product_{j>0} (1-x^(4*j)+x^(5*j))/(1-x^j).
%F A184639 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(-4*x) + exp(-6*x)) dx = 0.77101366877372033648945034346499691865027592089088481444183... - _Vaclav Kotesovec_, Jun 12 2025
%e A184639 a(4) = 4, because 4 partitions of 4 have no parts with multiplicity 4: [1,1,2], [2,2], [1,3], [4].
%p A184639 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
%p A184639       add((l->`if`(j=4, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
%p A184639     end:
%p A184639 a:= n-> (l-> l[1]-l[2])(b(n, n)):
%p A184639 seq(a(n), n=0..50);
%t A184639 b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i<1, {0, 0}, Sum[Function[l, If[j == 4, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
%t A184639 a[n_] := b[n, n][[1]] - b[n, n][[2]];
%t A184639 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)
%Y A184639 Cf. A000041, A183561, A183568, A007690, A116645, A118807, A184640, A184641, A184642, A184643, A184644, A184645.
%K A184639 nonn
%O A184639 0,3
%A A184639 _Alois P. Heinz_, Jan 18 2011