0,5

Last term in each row forms A107877, the number of subpartitions of the partition consisting of the triangular numbers.

To obtain row 4 from row 3:

[1, 3, _5, _7, _7, _7, __7];

take partial sums with final term '37' repeated 4 more times:

[1, 4, _9, 16, 23, 30, _37, _37, _37, _37, _37].

To obtain row 5, take partial sums of row 4 with the final term '268'

repeated 5 more times at the end:

[1, 5, 14, 30, 53, 83, 120, 157, 194, 231, 268, 268,268,268,268,268].

Triangle begins:

1;

1, 1;

1, 2, 2, 2;

1, 3, 5, 7, 7, 7, 7;

1, 4, 9, 16, 23, 30, 37, 37, 37, 37, 37;

1, 5, 14, 30, 53, 83, 120, 157, 194, 231, 268, 268, 268, 268, 268, 268;

1, 6, 20, 50, 103, 186, 306, 463, 657, 888, 1156, 1424, 1692, 1960, 2228, 2496, 2496, 2496, 2496, 2496, 2496, 2496;

Final term in rows forms A107877:

[1, 1, 2, 7, 37, 268, 2496, 28612, 391189, 6230646, 113521387, ...]

which satisfies the g.f.:

1/(1-x) = 1 + 1*x*(1-x) + 2*x^2*(1-x)^3 + 7*x^3*(1-x)^6 +

37*x^4*(1-x)^10 + 268*x^5*(1-x)^15 + 2496*x^6*(1-x)^21 +...

(PARI) {T(n, k)=if(n<0|k<0|k>n*(n+1)/2, 0, if(k==0, 1, if(k<=n*(n-1)/2, T(n, k-1)+T(n-1, k), T(n, k-1))))}

Cf. A107877 (leading edge); diagonals: A127497, A127498.

Sequence in context: A026268 A089258 A004065 this_sequence A144393 A089400 A105777

Adjacent sequences: A127493 A127494 A127495 this_sequence A127497 A127498 A127499

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Jan 16 2007