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%I #12 Mar 13 2021 17:29:00
%S 0,0,0,0,1,1,1,2,4,5,6,9,12,16,20,26,34,43,53,67,82,101,124,151,184,
%T 222,267,320,381,454,539,637,752,884,1038,1214,1417,1651,1920,2227,
%U 2578,2979,3437,3957,4547,5218,5980,6840,7815,8914,10154,11552,13122
%N Number of partitions of n with exactly one repeated part and that part is even.
%H Andrew Howroyd, <a href="/A341496/b341496.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 G.f.: (Sum_{k>=1} x^(4*k)/(1 - x^(4*k)) * Product_{k>=1} (1 + x^k).
%F a(n) = A090867(n) - A341497(n).
%F a(n) = A341497(n) - A116680(n).
%F a(n) = A341494(n) for even n; a(n) = A341495(n) for odd n.
%e The a(4) = 1 partition is: 2+2.
%e The a(5) = 1 partition is: 1+2+2.
%e The a(6) = 1 partition is: 2+2+2.
%e The a(7) = 2 partitions are: 2+2+3, 1+2+2+2.
%e The a(8) = 4 partitions are: 4+4, 2+2+4, 1+2+2+3, 2+2+2+2.
%o (PARI) seq(n)={Vec(sum(k=1, n\4, x^(4*k)/(1 - x^(4*k)) + O(x*x^n)) * prod(k=1, n, 1 + x^k + O(x*x^n)), -(n+1))}
%Y Cf. A090867, A116680, A341494, A341495, A341497.
%K nonn
%O 0,8
%A _Andrew Howroyd_, Feb 13 2021
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