login
Number of partitions of n into an even number of parts that are not multiples of 3.
4

%I #6 Dec 03 2020 12:18:51

%S 1,0,1,1,2,2,4,4,7,8,11,13,19,21,29,35,45,53,69,80,102,121,149,176,

%T 218,254,310,365,438,513,616,716,853,994,1172,1362,1604,1853,2170,

%U 2509,2920,3365,3909,4488,5193,5958,6862,7854,9030,10303,11809,13460,15376,17487,19941,22624,25736,29161

%N Number of partitions of n into an even number of parts that are not multiples of 3.

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F G.f.: (1/2) * (Product_{k>=1} (1 - x^(3*k)) / (1 - x^k) + Product_{k>=1} (1 + x^(3*k)) / (1 + x^k)).

%F a(n) = (A000726(n) + A109389(n)) / 2.

%e a(7) = 4 because we have [5, 2], [4, 1, 1, 1], [2, 2, 2, 1] and [2, 1, 1, 1, 1, 1].

%p b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,

%p b(n, i-1, t)+`if`(irem(i, 3)=0, 0, b(n-i, min(n-i, i), 1-t))))

%p end:

%p a:= n-> b(n$2, 1):

%p seq(a(n), n=0..60); # _Alois P. Heinz_, Dec 03 2020

%t nmax = 57; CoefficientList[Series[(1/2) (Product[(1 - x^(3 k))/(1 - x^k), {k, 1, nmax}] + Product[(1 + x^(3 k))/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

%Y Cf. A000726, A001651, A027187, A109389, A339405, A339406, A339407.

%K nonn

%O 0,5

%A _Ilya Gutkovskiy_, Dec 03 2020