T(n,m) tabf head (staircase) for  A097305: 

  Number of partitions of n with odd parts only and largest part k= 2*m-1, m=1..ceiling(n/2).

 
     \k       1         3         5         7         9        11        13        15        17

    n\m       1         2         3         4         5         6         7         8         9
  ________________________________________________________________________________________________
  
    1         1         0         0         0         0         0         0         0         0

    2         1         0         0         0         0         0         0         0         0

    3         1         1         0         0         0         0         0         0         0

    4         1         1         0         0         0         0         0         0         0

    5         1         1         1         0         0         0         0         0         0

    6         1         2         1         0         0         0         0         0         0

    7         1         2         1         1         0         0         0         0         0

    8         1         2         2         1         0         0         0         0         0

    9         1         3         2         1         1         0         0         0         0

   10         1         3         3         2         1         0         0         0         0
 
   11         1         3         4         2         1         1         0         0         0
 
   12         1         4         4         3         2         1         0         0         0
 
   13         1         4         5         4         2         1         1         0         0
 
   14         1         4         6         5         3         2         1         0         0
   
   15         1         5         7         6         4         2         1         1         0
   
   16         1         5         8         7         5         3         2         1         0
   
   17         1         5         9         9         6         4         2         1         1
   
   18         1         6        10        10         8         5         3         2         1

   etc.

  

  e.g.: T(10,3)= 3 for the partitions of n=10 with odd parts and largest part 5, viz. (5^2),(1^2,3,5), and (1^5,5).