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a(n) is the number of odd-balanced unimodal sequences of weight 2n+2.
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%I #10 Jun 22 2020 07:02:24

%S 1,2,5,9,16,29,48,77,123,191,290,436,643,936,1352,1927,2720,3810,5287,

%T 7282,9965,13539,18280,24545,32769,43519,57522,75667,99092,129237,

%U 167862,217192,280003,359695,460513,587702

%N a(n) is the number of odd-balanced unimodal sequences of weight 2n+2.

%C A unimodal sequence is odd-balanced if: (i) the peak is even and unique, (ii) even parts to the left of the peak are distinct, (iii) even parts to the right of the peak are distinct, (iv) odd parts to the left of the peak are identical to the odd parts to the right of the peak.

%H B. Kim, S. Lim, and J. Lovejoy, <a href="https://doi.org/10.1090/proc/13027">Odd-balanced unimodal sequences and related functions: parity, mock modularity and quantum modularity</a>, Proceedings of the American Mathematical Society, 144 (2016), 3687-3700.

%F G.f.: 1/(1-q) + Sum_{n>=1} q^n*(Product_{k=1..n} (1+q^k)^2)/(Product_{k=1..n+1} (1-q^(2*k-1))).

%e a(4) = 16, the relevant odd-balanced unimodal sequences being [10], [1,8,1], [8,2], [2,8], [1,1,6,1,1], [2,6,2], [4,6], [6,4], [1,6,2,1], [1,2,6,1], [1,1,1,4,1,1,1], [1,2,4,2,1], [1,1,2,4,1,1], [1,1,4,2,1,1], [3,4,3], [1,1,1,1,2,1,1,1,1].

%o (PARI) my(N=44, q='q+O('q^N)); Vec( 1/(1-q) + sum(n=1, N, q^n * prod(k=1,n, (1+q^k)^2) / prod(k=1,n+1, 1-q^(2*k-1)) ) ) \\ _Joerg Arndt_, Jun 22 2020

%Y Cf. A001523, A059618.

%K nonn

%O 0,2

%A _Jeremy Lovejoy_, Jun 21 2020