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%I #11 May 10 2022 14:38:10
%S 0,0,0,1,1,2,3,6,8,13,18,26,36,51,68,93,123,164,214,278,358,461,586,
%T 744,937,1177,1468,1826,2260,2788,3425,4195,5118,6228,7552,9135,11016,
%U 13252,15899,19031,22724,27075,32188,38189,45214,53431,63020,74200,87205,102318
%N Number of partitions of n in which exactly one even part is repeated and odd parts are unrestricted.
%H George E. Andrews and Mircea Merca, <a href="https://doi.org/10.1007/s00026-019-00479-y">On the Number of Even Parts in All Partitions of n into Distinct Parts</a>, Ann. Comb. 24, 47-54 (2020).
%H Nyirenda Darlison and Mugwangwavari Beaullah, <a href="https://arxiv.org/abs/2205.03697">Extentions and variations of Andrews-Merca identities</a>, arXiv:2205.03697 [math.CO], 2022. See p. 7 ff.
%H Shishuo Fu and Dazhao Tang, <a href="https://arxiv.org/abs/1705.05046">Generalizing a partition theorem of Andrews</a>, arXiv:1705.05046 [math.CO], 2017.
%o (PARI) a(n) = {my(nbp=0); forpart(p=n, my(s=Set(p), nber=0, vp=Vec(p)); for (i=1, #s, if (((s[i] % 2) == 0) && (#select(x->(x==s[i]), vp) > 1), nber++);); if (nber == 1, nbp++);); nbp;}
%Y Cf. A000041, A353903.
%K nonn
%O 1,6
%A _Michel Marcus_, May 10 2022