cp's OEIS Frontend

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.

%I A335262 #20 May 30 2020 05:06:23
%S A335262 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 A335262 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 A335262 21,10,3,0,1,6,15,28,45,66,45,28,15,6,1,0,3,10,21,36,55
%N 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.
%F A335262 T(n, k) = Pochhammer(2*k - 1 - n, 2) / 2!.
%F A335262 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 A335262 T(n, k) = (2*k-1-n)*(2*k-n)/2. - _Michel Marcus_, May 29 2020
%e A335262 Triangle starts:
%e A335262                                0;
%e A335262                               1, 0;
%e A335262                             3, 0, 1;
%e A335262                            6, 1, 0, 3;
%e A335262                          10, 3, 0, 1, 6;
%e A335262                        15, 6, 1, 0, 3, 10;
%e A335262                      21, 10, 3, 0, 1, 6, 15;
%e A335262                    28, 15, 6, 1, 0, 3, 10, 21;
%e A335262                  36, 21, 10, 3, 0, 1, 6, 15, 28;
%e A335262                45, 28, 15, 6, 1, 0, 3, 10, 21, 36;
%e A335262              55, 36, 21, 10, 3, 0, 1, 6, 15, 28, 45;
%e A335262            66, 45, 28, 15, 6, 1, 0, 3, 10, 21, 36, 55;
%e A335262          78, 55, 36, 21, 10, 3, 0, 1, 6, 15, 28, 45, 66;
%p A335262 T := (n,k) -> pochhammer(2*k - 1 - n, 2)/2:
%p A335262 seq(seq(T(n,k), k=0..n), n=0..11);
%o A335262 (Python)
%o A335262 def T(num_rows):
%o A335262     t, s = 1, 1
%o A335262     L, R = [0], [0]
%o A335262     for n in range(1, num_rows):
%o A335262         R.reverse()
%o A335262         R.insert(0, t)
%o A335262         L.extend(R)
%o A335262         t, s = t+s+1, s+1
%o A335262     return L
%o A335262 print(T(12))
%o A335262 (PARI) T(n, k) = (2*k-1-n)*(2*k-n)/2;
%o A335262 tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, May 29 2020
%Y A335262 Row sums give the triangular pyramidal numbers A000292.
%Y A335262 Cf. A000217 (triangular numbers), A112367, A181940.
%K A335262 nonn,tabl
%O A335262 0,4
%A A335262 _Peter Luschny_, May 29 2020