A161039 - OEIS

%I #16 Jan 05 2016 12:48:51

%S 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,

%T 18,25,22,25,32,29,31,42,41,41,53,51,54,69,64,69,88,83,89,109,105,112,

%U 136,134,141,170,166,177,215,207,219,262,260,276,320,320,341,397,397,417,485

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

%H R. H. Hardin and Vaclav Kotesovec, <a href="/A161039/b161039.txt">Table of n, a(n) for n = 1..5000</a> (first 1000 terms from R. H. Hardin)

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

%F 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

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

%p 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

%t 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 *)

%Y Cf. A100405.

%K nonn

%O 1,9

%A _R. H. Hardin_ Jun 02 2009

%E Minor edits by _Vaclav Kotesovec_, Jan 02 2016