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A161294 Number of partitions of n into numbers not divisible by 4 where every part appears at least 3 times.

%I A161294 #17 Jun 15 2025 10:40:12
%S A161294 0,0,1,1,1,2,1,2,3,3,3,6,5,6,10,8,9,15,13,16,22,20,24,33,32,36,47,48,
%T A161294 53,71,68,77,100,99,112,140,138,158,194,199,219,268,275,305,369,377,
%U A161294 416,501,514,572,671,693,768,898,935,1028,1189,1245,1364,1576,1642,1798,2063
%N A161294 Number of partitions of n into numbers not divisible by 4 where every part appears at least 3 times.
%H A161294 R. H. Hardin, <a href="/A161294/b161294.txt">Table of n, a(n) for n = 1..1000</a>
%F A161294 G.f.: -1 + (Product_{j>=1} (1 + x^(3*j)/(1-x^j)))/Product_{j>=1} (1 + x^(12*j)/(1-x^(4*j))). - _Emeric Deutsch_, Jun 21 2009
%F A161294 a(n) ~ (6*c + Pi^2)^(1/4) * exp(sqrt((6*c + Pi^2)*n/2)) / (2^(11/4) * sqrt(Pi) * n^(3/4)), where c = Integral_{x=0..oo} log(1 - exp(-x) + exp(-3*x)) dx = -0.77271248407593487127235205445116662610863126869049971822566... - _Vaclav Kotesovec_, Jun 15 2025
%e A161294 a(13)=5 because we have (3^3)(1^4), (2^5)(1^3), (2^4)(1^5), (2^3)(1^7), and 1^(13). - _Emeric Deutsch_, Jun 21 2009
%p A161294 g := -1+(product(1+x^(3*j)/(1-x^j), j = 1 .. 40))/(product(1+x^(12*j)/(1-x^(4*j)), j = 1 .. 40)): gser := series(g, x = 0, 70): seq(coeff(gser, x, n), n = 2 .. 68); # _Emeric Deutsch_, Jun 21 2009
%t A161294 (* Closed form for the constant c: *) N[Pi^2/3 + 1/2*(Log[-1/3 - 1/6*(1 + I*Sqrt[3])*(25/2 - (3*Sqrt[69])/2)^(1/3) - 1/6*(1 - I*Sqrt[3])*(1/2*(25 + 3*Sqrt[69]))^(1/3)]^2 + Log[-1/3 - 1/6*(1 - I*Sqrt[3])*(25/2 - (3*Sqrt[69])/2)^(1/3) - 1/6*(1 + I*Sqrt[3])*((1/2)*(25 + 3*Sqrt[69]))^(1/3)]^2 + 2*(-PolyLog[2, 1/3*(1 - (2/(25 - 3*Sqrt[69]))^(1/3) - (1/2*(25 - 3*Sqrt[69]))^(1/3))] + PolyLog[2, ((1 + I*Sqrt[3])*(1/2*(9 - Sqrt[69]))^(1/3))/(2*3^(2/3)) + (1 - I*Sqrt[3])/(2^(2/3)*(3*(9 - Sqrt[69]))^(1/3))] + PolyLog[2, ((1 - I*Sqrt[3])*(1/2*(9 - Sqrt[69]))^(1/3)) / (2*3^(2/3)) + (1 + I*Sqrt[3])/(2^(2/3)*(3*(9 - Sqrt[69]))^(1/3))])), 100] // Chop (* _Vaclav Kotesovec_, Jun 15 2025 *)
%Y A161294 Cf. A100405.
%K A161294 nonn
%O A161294 1,6
%A A161294 _R. H. Hardin_, Jun 06 2009