cp's OEIS Frontend

A339407 Number of partitions of n into an odd number of parts that are not multiples of 4.

%I A339407 #9 Nov 23 2021 09:43:54
%S A339407 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 A339407 341,459,511,664,737,957,1073,1357,1518,1918,2156,2673,3002,3712,4182,
%U A339407 5100,5737,6976,7866,9460,10652,12777,14402,17126,19284,22867,25761,30340,34139,40099
%N A339407 Number of partitions of n into an odd number of parts that are not multiples of 4.
%H A339407 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 A339407 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A339407 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 A339407 a(n) = (A001935(n) - A261734(n)) / 2.
%e A339407 a(6) = 4 because we have [6], [3, 2, 1], [2, 2, 2] and [2, 1, 1, 1, 1].
%p A339407 b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
%p A339407       b(n, i-1, t)+`if`(irem(i, 4)=0, 0, b(n-i, min(n-i, i), 1-t))))
%p A339407     end:
%p A339407 a:= n-> b(n$2, 0):
%p A339407 seq(a(n), n=0..55);  # _Alois P. Heinz_, Dec 03 2020
%t A339407 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 A339407 Cf. A001935, A027193, A042968, A261734, A339404, A339405, A339406.
%K A339407 nonn
%O A339407 0,4
%A A339407 _Ilya Gutkovskiy_, Dec 03 2020