A335262 Triangle of triangular numbers, read by rows, constructed like this: Given a sequence t, start row 0 with t(0). Compute row n for n > 0 by reversing row n-1 and prepending t(n). The sequence t is here chosen as the triangular numbers.
0, 1, 0, 3, 0, 1, 6, 1, 0, 3, 10, 3, 0, 1, 6, 15, 6, 1, 0, 3, 10, 21, 10, 3, 0, 1, 6, 15, 28, 15, 6, 1, 0, 3, 10, 21, 36, 21, 10, 3, 0, 1, 6, 15, 28, 45, 28, 15, 6, 1, 0, 3, 10, 21, 36, 55, 36, 21, 10, 3, 0, 1, 6, 15, 28, 45, 66, 45, 28, 15, 6, 1, 0, 3, 10, 21, 36, 55
Offset: 0
Examples
Triangle starts:
0;
1, 0;
3, 0, 1;
6, 1, 0, 3;
10, 3, 0, 1, 6;
15, 6, 1, 0, 3, 10;
21, 10, 3, 0, 1, 6, 15;
28, 15, 6, 1, 0, 3, 10, 21;
36, 21, 10, 3, 0, 1, 6, 15, 28;
45, 28, 15, 6, 1, 0, 3, 10, 21, 36;
55, 36, 21, 10, 3, 0, 1, 6, 15, 28, 45;
66, 45, 28, 15, 6, 1, 0, 3, 10, 21, 36, 55;
78, 55, 36, 21, 10, 3, 0, 1, 6, 15, 28, 45, 66;
Crossrefs
Programs
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Maple
T := (n,k) -> pochhammer(2*k - 1 - n, 2)/2: seq(seq(T(n,k), k=0..n), n=0..11);
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PARI
T(n, k) = (2*k-1-n)*(2*k-n)/2; tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 29 2020
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Python
def T(num_rows): t, s = 1, 1 L, R = [0], [0] for n in range(1, num_rows): R.reverse() R.insert(0, t) L.extend(R) t, s = t+s+1, s+1 return L print(T(12))
Formula
T(n, k) = Pochhammer(2*k - 1 - n, 2) / 2!.
Row n is generated by the quadratic polynomial 2*x^2 - (2*n+5)*x + t(n+2), where t(n) are the triangular numbers, evaluated at x = k + 1.
T(n, k) = (2*k-1-n)*(2*k-n)/2. - Michel Marcus, May 29 2020