A127496 Triangle, read by rows of n*(n+1)/2 + 1 terms, generated by the following rule: start with a single '1' in row n=0; subsequently, row n+1 equals the partial sums of row n with the final term repeated n+1 more times at the end.

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

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

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

Cf. A305605, A305601.

Sequence in context: A026268 A089258 A004065 * A350189 A289778 A277523

Adjacent sequences: A127493 A127494 A127495 * A127497 A127498 A127499

Keyword

nonn,tabl

Author

Paul D. Hanna, Jan 16 2007

Status

approved