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A265254 Number of partitions of n having no even singletons.

%I A265254 #32 Mar 21 2025 02:20:31
%S A265254 1,1,1,2,3,4,6,8,11,15,19,25,34,43,54,70,89,111,140,174,216,268,328,
%T A265254 402,495,601,727,883,1066,1281,1540,1843,2202,2627,3120,3702,4392,
%U A265254 5187,6114,7206,8471,9936,11644,13617,15902,18548,21588,25098,29156,33799,39129
%N A265254 Number of partitions of n having no even singletons.
%H A265254 Alois P. Heinz, <a href="/A265254/b265254.txt">Table of n, a(n) for n = 0..5000</a>
%H A265254 Cristina Ballantine and Amanda Welch, <a href="https://arxiv.org/abs/2308.06136">Generalizations of POD and PED partitions</a>, arXiv:2308.06136 [math.CO], 2023. See pp. 15-16.
%H A265254 James A. Sellers, <a href="https://arxiv.org/abs/2308.09999">Elementary Proofs of Congruences for POND and PEND Partitions</a>, arXiv:2308.09999 [math.NT], 2023.
%H A265254 James A. Sellers and Nicolas Allen Smoot, <a href="https://arxiv.org/abs/2503.16019">Explaining Unforeseen Congruence Relationships Between PEND and POND Partitions via an Atkin--Lehner Involution</a>, arXiv:2503.16019 [math.NT], 2025.
%F A265254 G.f.: g(x) = Product_{j>=1} (1 - x^(2j) + x^(4j))/(1-x^j).
%F A265254 a(n) = A265253(n,0).
%F A265254 G.f.: Product_{k>=1} (1 + x^k) * (1 + x^(6*k)) / (1 - x^(4*k)). - _Vaclav Kotesovec_, Jan 01 2016
%F A265254 a(n) ~ sqrt(5) * exp(sqrt(5*n)*Pi/3) / (3*2^(5/2)*n). - _Vaclav Kotesovec_, Jan 01 2016
%e A265254 a(5) = 4 because the partitions [1,1,1,1,1], [1,2,2], [1,1,3], [5] have no even singletons while [1,1,1,2], [2,3], [1,4] do have.
%p A265254 g := mul((1-x^(2*j)+x^(4*j))/(1-x^j), j = 1 .. 80): gser := series(g, x = 0,65): seq(coeff(gser, x, n), n = 0 .. 60);
%p A265254 # second Maple program:
%p A265254 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A265254       `if`(j=1 and i::even, 0, b(n-i*j, i-1)), j=0..n/i)))
%p A265254     end:
%p A265254 a:= n-> b(n$2):
%p A265254 seq(a(n), n=0..60);  # _Alois P. Heinz_, Jan 02 2016
%t A265254 nmax = 50; CoefficientList[Series[Product[(1 - x^(2*k) + x^(4*k)) / (1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 01 2016 *)
%t A265254 nmax = 50; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(6*k)) / (1 - x^(4*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 01 2016 *)
%Y A265254 Cf. A265253.
%K A265254 nonn
%O A265254 0,4
%A A265254 _Emeric Deutsch_, Dec 31 2015