A161039 Number of partitions of n into odd numbers where every part appears at least 3 times.

0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 2, 5, 3, 3, 6, 5, 6, 8, 6, 7, 11, 10, 9, 14, 13, 13, 19, 16, 18, 25, 22, 25, 32, 29, 31, 42, 41, 41, 53, 51, 54, 69, 64, 69, 88, 83, 89, 109, 105, 112, 136, 134, 141, 170, 166, 177, 215, 207, 219, 262, 260, 276, 320, 320, 341, 397, 397, 417, 485

Offset: 1

Author

R. H. Hardin Jun 02 2009

Keywords

• nonn

Examples

			a(15)=5 because we have 333, (2^6)(1^3), (2^5)(1^5), (2^4)(1^7), and (2^3)(1^9).
		

Links

• R. H. Hardin and Vaclav Kotesovec, Table of n, a(n) for n = 1..5000 (first 1000 terms from R. H. Hardin)

Crossrefs

Cf. A100405.

Programs

Maple

g := product(1+x^(3*(2*j-1))/(1-x^(2*j-1)), j = 1 .. 20): gser := series(g, x = 0, 80): seq(coeff(gser, x, n), n = 1 .. 72); # Emeric Deutsch, Jun 26 2009

Mathematica

nmax = 100; Rest[CoefficientList[Series[Product[1 + x^(6*k-3) / (1-x^(2*k-1)), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 02 2016 *)

Formula

G.f.: Product_{j>=1} (1 + x^(6j-3)/(1-x^(2j-1))). - Emeric Deutsch, Jun 26 2009

a(n) ~ (6*c + Pi^2)^(1/4) * exp(sqrt((6*c + Pi^2)*n/3)) / (4*3^(1/4)*sqrt(Pi) * n^(3/4)), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-3*x)) dx = -0.77271248407593487127235205445116662610863126869049971822566... . - Vaclav Kotesovec, Jan 05 2016

Extensions

Minor edits by Vaclav Kotesovec, Jan 02 2016