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A100405 Number of partitions of n where every part appears more than two times.

%I A100405 #25 Jun 15 2025 10:39:20
%S A100405 1,0,0,1,1,1,2,1,2,3,3,3,7,5,6,11,10,10,17,15,20,26,25,29,44,41,47,63,
%T A100405 67,72,99,97,114,143,148,168,216,216,248,306,328,358,443,462,527,629,
%U A100405 665,739,898,936,1055,1238,1330,1465,1727,1837,2055,2366,2543,2808,3274
%N A100405 Number of partitions of n where every part appears more than two times.
%H A100405 Alois P. Heinz, <a href="/A100405/b100405.txt">Table of n, a(n) for n = 0..10000</a> (terms n = 1..5000 from Vaclav Kotesovec)
%F A100405 G.f.: Product_{k>0} (1+x^(3*k)/(1-x^k)). More generally, g.f. for number of partitions of n where every part appears more than m times is Product_{k>0} (1+x^((m+1)*k)/(1-x^k)).
%F A100405 a(n) ~ sqrt(Pi^2 + 6*c) * exp(sqrt((2*Pi^2/3 + 4*c)*n)) / (4*sqrt(3)*Pi*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-3*x)) dx = -0.77271248407593487127235205445116662610863126869049971822566... . - _Vaclav Kotesovec_, Jan 05 2016
%e A100405 a(6)=2 because we have [2,2,2] and [1,1,1,1,1,1].
%p A100405 G:=product((1+x^(3*k)/(1-x^k)),k=1..30): Gser:=series(G,x=0,80): seq(coeff(Gser,x,n),n=0..70); # _Emeric Deutsch_, Aug 06 2005
%p A100405 # second Maple program:
%p A100405 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A100405       add(b(n-i*j, i-1), j=[0, $3..iquo(n, i)])))
%p A100405     end:
%p A100405 a:= n-> b(n$2):
%p A100405 seq(a(n), n=0..70);  # _Alois P. Heinz_, Aug 20 2019
%t A100405 nmax = 100; Rest[CoefficientList[Series[Product[1 + x^(3*k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Nov 28 2015 *)
%Y A100405 Cf. A007690, A160974-A160990.
%Y A100405 Cf. A266647, A266648, A266649, A266650.
%Y A100405 Cf. A161294.
%K A100405 easy,nonn
%O A100405 0,7
%A A100405 _Vladeta Jovovic_, Jan 11 2005
%E A100405 More terms from _Emeric Deutsch_, Aug 06 2005
%E A100405 a(0)=1 prepended by _Alois P. Heinz_, Aug 20 2019