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

0,4

Links

Table of n, a(n) for n=0..77.

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

Example

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;

Maple

T := (n, k) -> pochhammer(2*k - 1 - n, 2)/2:
seq(seq(T(n, k), k=0..n), n=0..11);

Prog

(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))
(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

Crossrefs

Row sums give the triangular pyramidal numbers A000292.

Cf. A000217 (triangular numbers), A112367, A181940.

Sequence in context: A129684 A247255 A105147 * A111924 A212880 A211510

Adjacent sequences: A335259 A335260 A335261 * A335263 A335264 A335265

Keyword

nonn,tabl

Author

Peter Luschny, May 29 2020

Status

approved