OFFSET
0,5
COMMENTS
Last term in each row forms A107877, the number of subpartitions of the partition consisting of the triangular numbers.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
EXAMPLE
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 +...
MATHEMATICA
nxt[h_] :=Module[{c = Accumulate[h]}, Join[c, PadRight[{}, c[[2]], c[[-1]]]]]; Join[{1}, Flatten[NestList[nxt, {1, 1}, 5]]] (* Harvey P. Dale, Mar 10 2020 *)
PROG
(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))))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jan 16 2007
STATUS
approved