%I #20 May 30 2020 05:06:23
%S 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,
%T 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,
%U 21,10,3,0,1,6,15,28,45,66,45,28,15,6,1,0,3,10,21,36,55
%N 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.
%F T(n, k) = Pochhammer(2*k - 1 - n, 2) / 2!.
%F 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.
%F T(n, k) = (2*k-1-n)*(2*k-n)/2. - _Michel Marcus_, May 29 2020
%e Triangle starts:
%e 0;
%e 1, 0;
%e 3, 0, 1;
%e 6, 1, 0, 3;
%e 10, 3, 0, 1, 6;
%e 15, 6, 1, 0, 3, 10;
%e 21, 10, 3, 0, 1, 6, 15;
%e 28, 15, 6, 1, 0, 3, 10, 21;
%e 36, 21, 10, 3, 0, 1, 6, 15, 28;
%e 45, 28, 15, 6, 1, 0, 3, 10, 21, 36;
%e 55, 36, 21, 10, 3, 0, 1, 6, 15, 28, 45;
%e 66, 45, 28, 15, 6, 1, 0, 3, 10, 21, 36, 55;
%e 78, 55, 36, 21, 10, 3, 0, 1, 6, 15, 28, 45, 66;
%p T := (n,k) -> pochhammer(2*k - 1 - n, 2)/2:
%p seq(seq(T(n,k), k=0..n), n=0..11);
%o (Python)
%o def T(num_rows):
%o t, s = 1, 1
%o L, R = [0], [0]
%o for n in range(1, num_rows):
%o R.reverse()
%o R.insert(0, t)
%o L.extend(R)
%o t, s = t+s+1, s+1
%o return L
%o print(T(12))
%o (PARI) T(n, k) = (2*k-1-n)*(2*k-n)/2;
%o tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, May 29 2020
%Y Row sums give the triangular pyramidal numbers A000292.
%Y Cf. A000217 (triangular numbers), A112367, A181940.
%K nonn,tabl
%O 0,4
%A _Peter Luschny_, May 29 2020