A339407 - OEIS

%I #9 Nov 23 2021 09:43:54

%S 0,1,1,2,1,4,4,7,6,13,13,21,21,36,38,57,59,90,98,137,148,210,231,310,

%T 341,459,511,664,737,957,1073,1357,1518,1918,2156,2673,3002,3712,4182,

%U 5100,5737,6976,7866,9460,10652,12777,14402,17126,19284,22867,25761,30340,34139,40099

%N Number of partitions of n into an odd number of parts that are not multiples of 4.

%H Cristina Ballantine and Mircea Merca, <a href="https://arxiv.org/abs/2111.10702">4-Regular partitions and the pod function</a>, arXiv:2111.10702 [math.CO], 2021.

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

%F G.f.: (1/2) * (Product_{k>=1} (1 - x^(4*k)) / (1 - x^k) - Product_{k>=1} (1 + x^(4*k)) / (1 + x^k)).

%F a(n) = (A001935(n) - A261734(n)) / 2.

%e a(6) = 4 because we have [6], [3, 2, 1], [2, 2, 2] and [2, 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, 4)=0, 0, b(n-i, min(n-i, i), 1-t))))

%p     end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..55);  # _Alois P. Heinz_, Dec 03 2020

%t nmax = 53; CoefficientList[Series[(1/2) (Product[(1 - x^(4 k))/(1 - x^k), {k, 1, nmax}] - Product[(1 + x^(4 k))/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

%Y Cf. A001935, A027193, A042968, A261734, A339404, A339405, A339406.

%K nonn

%O 0,4

%A _Ilya Gutkovskiy_, Dec 03 2020