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A341494 Number of partitions of n into an even number of parts such that the set of even parts has only one element.

%I A341494 #17 May 24 2021 11:54:49
%S A341494 0,0,0,1,1,3,1,7,4,13,6,23,12,39,20,63,34,98,53,150,82,225,124,329,
%T A341494 184,475,267,676,381,948,539,1317,752,1810,1038,2460,1417,3319,1920,
%U A341494 4442,2578,5897,3437,7780,4547,10200,5980,13285,7815,17214,10154,22191,13122
%N A341494 Number of partitions of n into an even number of parts such that the set of even parts has only one element.
%H A341494 Andrew Howroyd, <a href="/A341494/b341494.txt">Table of n, a(n) for n = 0..1000</a>
%H A341494 Cristina Ballantine and Mircea Merca, <a href="https://doi.org/10.1007/s11139-019-00184-7">Combinatorial proofs of two theorems related to the number of even parts in all partitions of n into distinct parts</a>, Ramanujan J.,  54:1 (2021), 107-112.
%F A341494 G.f.: (P(x,1) + P(x,-1))/2 where P(x,c) = (Sum_{k>=1} c*x^(2*k)/(1-c*x^(2*k))) / (Product_{k>=1} 1-c*x^(2*k-1)).
%F A341494 a(n) = A090867(n) - A341495(n).
%e A341494 The a(3) = 1 partition is: 1+2.
%e A341494 The a(4) = 1 partition is: 2+2.
%e A341494 The a(5) = 3 partitions are: 1+4, 2+3, 1+1+1+2.
%t A341494 P[n_, c_] := c*Sum[x^(2k)/(1 - c*x^(2k)) + O[x]^n, {k, 1, n/2}]/
%t A341494      Product[1 - c*x^(2k - 1) + O[x]^n, {k, 1, n/2}];
%t A341494 CoefficientList[(P[100, 1] + P[100, -1])/2, x] (* _Jean-François Alcover_, May 24 2021, from PARI code *)
%o A341494 (PARI)
%o A341494 P(n,c)={c*sum(k=1, n\2, x^(2*k)/(1-c*x^(2*k)) + O(x*x^n))/prod(k=1, n\2, 1-c*x^(2*k-1) + O(x*x^n))}
%o A341494 seq(n)={Vec(P(n,1) + P(n,-1), -(n+1))/2}
%Y A341494 Cf. A090867, A341495, A341496, A341497.
%K A341494 nonn
%O A341494 0,6
%A A341494 _Andrew Howroyd_, Feb 13 2021