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A265256 Number of partitions of n having no odd singletons (n>=0).

%I A265256 #22 Mar 21 2025 02:20:35
%S A265256 1,0,2,1,4,2,8,4,14,9,24,16,41,28,66,49,104,80,163,128,248,203,372,
%T A265256 312,554,472,810,708,1172,1042,1684,1516,2390,2188,3364,3118,4705,
%U A265256 4404,6522,6177,8980,8584,12295,11844,16718,16244,22604,22120,30413,29944,40692
%N A265256 Number of partitions of n having no odd singletons (n>=0).
%H A265256 Alois P. Heinz, <a href="/A265256/b265256.txt">Table of n, a(n) for n = 0..5000</a>
%H A265256 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 A265256 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 A265256 a(n) = A265255(n,0).
%F A265256 G.f.: g(x) = Product_{j>=1} (1 - x^(2j-1) + x^(4j-2)) / (1-x^j).
%F A265256 a(n) ~ sqrt(5) * exp(sqrt(5*n)*Pi/3) / (12*sqrt(2)*n). - _Vaclav Kotesovec_, Jan 01 2016
%e A265256 a(5) = 2 because  among the 7 partitions of 5 only [1,1,1,1,1] and [1,1,1,2] have no odd singletons (the others are: [1,2,2], [1,1,3], [2,3], [1,4], [5]).
%p A265256 g := mul((1-x^(2*j-1)+x^(4*j-2))/(1-x^j), j = 1 .. 80): gser := series(g, x = 0, 60): seq(coeff(gser, x, n), n = 0 .. 55);
%p A265256 # second Maple program:
%p A265256 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A265256       `if`(j=1 and i::odd, 0, b(n-i*j, i-1)), j=0..n/i)))
%p A265256     end:
%p A265256 a:= n-> b(n$2):
%p A265256 seq(a(n), n=0..60);  # _Alois P. Heinz_, Jan 02 2016
%t A265256 nmax = 50; CoefficientList[Series[Product[((1 - x^(2*k-1) + x^(4*k-2))) / (1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 01 2016 *)
%t A265256 nmax = 50; CoefficientList[Series[Product[(1 + x^(6*k-3)) / ((1 + x^(2*k-1)) * (1-x^k)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 01 2016 *)
%Y A265256 Cf. A265254, A265255.
%K A265256 nonn
%O A265256 0,3
%A A265256 _Emeric Deutsch_, Jan 01 2016