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A339406 Number of partitions of n into an even number of parts that are not multiples of 4.

%I A339406 #10 Nov 23 2021 09:43:41
%S A339406 1,0,1,1,3,2,5,5,10,9,16,17,29,28,44,48,73,76,110,121,172,185,253,282,
%T A339406 381,417,549,616,802,889,1137,1279,1620,1810,2260,2549,3161,3544,4346,
%U A339406 4906,5979,6720,8120,9164,11014,12392,14788,16682,19820,22297,26337,29682,34921,39267
%N A339406 Number of partitions of n into an even number of parts that are not multiples of 4.
%H A339406 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 A339406 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A339406 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 A339406 a(n) = (A001935(n) + A261734(n)) / 2.
%e A339406 a(6) = 5 because we have [5, 1], [3, 3], [3, 1, 1, 1], [2, 2, 1, 1] and [1, 1, 1, 1, 1, 1].
%p A339406 b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
%p A339406       b(n, i-1, t)+`if`(irem(i, 4)=0, 0, b(n-i, min(n-i, i), 1-t))))
%p A339406     end:
%p A339406 a:= n-> b(n$2, 1):
%p A339406 seq(a(n), n=0..55);  # _Alois P. Heinz_, Dec 03 2020
%t A339406 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 A339406 Cf. A001935, A027187, A042968, A261734, A339404, A339405, A339407.
%K A339406 nonn
%O A339406 0,5
%A A339406 _Ilya Gutkovskiy_, Dec 03 2020