A generating function for triangular partitions

Abstract

Let T k ( n ) {T_k}(n) denote the number of solutions in nonnegative integers a i {a_i} , of the equation \[ n = ∑ i = 1 k ∑ j = 1 k − i + 1 a i j n = \sum \limits _{i = 1}^k {\sum \limits _{j = 1}^{k - i + 1} {{a_{ij}}} } \] where the a i j {a_{ij}} satisfy the inequalities a i j ⩾ a i + 1 , j , a i j ⩾ a i + 1 , j − 1 {a_{ij}} \geqslant {a_{i + 1,j}},{a_{ij}} \geqslant {a_{i + 1,j - 1}} . We show that \[ ∑ n = 1 ∞ T k ( n ) x n = ( 1 − x ) − k ( 1 − x 3 ) − k + 1 ( 1 − x 5 ) − k + 2 ⋯ ( 1 − x 2 k − 1 ) − 1 . \sum \limits _{n = 1}^\infty {{T_k}(n){x^n} = {{(1 - x)}^{ - k}}{{(1 - {x^3})}^{ - k + 1}}{{(1 - {x^5})}^{ - k + 2}} \cdots {{(1 - {x^{2k - 1}})}^{ - 1}}.} \]