A335768 a(n) is the number of odd-balanced unimodal sequences of weight 2n+2.

1, 2, 5, 9, 16, 29, 48, 77, 123, 191, 290, 436, 643, 936, 1352, 1927, 2720, 3810, 5287, 7282, 9965, 13539, 18280, 24545, 32769, 43519, 57522, 75667, 99092, 129237, 167862, 217192, 280003, 359695, 460513, 587702

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

Table of n, a(n) for n=0..35.

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

Cf. A001523, A059618.

Sequence in context: A326507 A014739 A290464 * A039946 A130752 A309331

Adjacent sequences: A335765 A335766 A335767 * A335769 A335770 A335771

Keyword

nonn

Author

Jeremy Lovejoy, Jun 21 2020

Status

approved