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

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, 53, 71, 68, 77, 100, 99, 112, 140, 138, 158, 194, 199, 219, 268, 275, 305, 369, 377, 416, 501, 514, 572, 671, 693, 768, 898, 935, 1028, 1189, 1245, 1364, 1576, 1642, 1798, 2063

Offset: 1

Author

R. H. Hardin, Jun 06 2009

Keywords

• nonn

Examples

			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
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A100405.

Programs

Maple

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

Mathematica

(* 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 *)

Formula

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

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