OFFSET
0,2
COMMENTS
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.
LINKS
B. Kim, S. Lim, and J. Lovejoy, Odd-balanced unimodal sequences and related functions: parity, mock modularity and quantum modularity, Proceedings of the American Mathematical Society, 144 (2016), 3687-3700.
FORMULA
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))).
EXAMPLE
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].
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Jeremy Lovejoy, Jun 21 2020
STATUS
approved