a(n,m) tabf head (staircase) for A103922 Rioran array S_n(m) with extra n=0 row added. See quoted Riordan ref, p.199, table. n\m 0 1 2 3 4 5 0 1 0 0 0 0 0 1 1 1 0 0 0 0 2 2 2 0 0 0 0 3 3 4 1 0 0 0 4 5 7 2 0 0 0 5 7 12 5 0 0 0 6 11 19 9 1 0 0 7 15 30 17 2 0 0 8 22 45 28 5 0 0 9 30 67 47 10 0 0 10 42 97 73 19 1 0 11 56 139 114 33 2 0 12 77 195 170 57 5 0 13 101 272 253 92 10 0 14 135 373 365 147 20 0 15 176 508 525 227 35 1 16 231 684 738 345 62 2 17 297 915 1033 512 102 5 18 385 1212 1422 752 167 10 19 490 1597 1948 1083 262 20 20 627 2087 2634 1545 407 36 . . . The length of row n is floor(1/2 + sqrt(2*(n+1))), n>=0. This is the sequence A002024(n+1)=[1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,...]. The column sequences are m=0: A000041 (partition nubers), m=1: A000070 (partial sums of partition numbers), m=2: A000097, m=3: A000098, m=4: A000710, m=5: A103924, m=6: A103925, m=7: A103926, m=8: A103927, m=9: A103928, m=10: A103929. The columns m>=0 without leading zeros are also the columns in the array p_{2}(n,m) of Gupta et al. (see ref. given in e.g. A103929) pp. 90, 91.