A368045 Triangle read by rows. T(n, k) = (k*(k + 1)*(2*k + 1) + n*(n + 1)*(2*n + 1)) / 6.

0, 1, 2, 5, 6, 10, 14, 15, 19, 28, 30, 31, 35, 44, 60, 55, 56, 60, 69, 85, 110, 91, 92, 96, 105, 121, 146, 182, 140, 141, 145, 154, 170, 195, 231, 280, 204, 205, 209, 218, 234, 259, 295, 344, 408, 285, 286, 290, 299, 315, 340, 376, 425, 489, 570

Offset: 0

Author

Peter Luschny, Dec 09 2023

Keywords

• nonn
• tabl
• easy

Comments

Consider a sequence-to-triangle transformation a -> T, where a is a 0-based sequence and T a regular (0, 0)-based triangular array. The transformation is recursively defined, starting with T(0, 0) = 0, and T(n, n) = a(n) + T(n, n - 1) for n > 0. For k <> n let T(n, k) = a(n) + T(n-1, k).

If a(n) = 1, then T = A051162; if a(n) = n, then T = A367964 (generalizing the triangular numbers); if a(n) = n^2, then T is this triangle.

In the multiplicative form of the transformation, T(0, 0) is set to 1, and the operation '+' is replaced by '*'. For instance, a(n) = 2 is then mapped to T = A368043 and a(n) = n to A143216.

Examples

			Triangle T(n, k) starts:
  [0] [  0]
  [1] [  1,   2]
  [2] [  5,   6,  10]
  [3] [ 14,  15,  19,  28]
  [4] [ 30,  31,  35,  44,  60]
  [5] [ 55,  56,  60,  69,  85, 110]
  [6] [ 91,  92,  96, 105, 121, 146, 182]
  [7] [140, 141, 145, 154, 170, 195, 231, 280]
  [8] [204, 205, 209, 218, 234, 259, 295, 344, 408]
  [9] [285, 286, 290, 299, 315, 340, 376, 425, 489, 570]
		

Crossrefs

Cf. A000330 (T(n,0)), A056520 (T(n,1)), A005900 (T(n-1,n)), A006331 (T(n,n)), A094952 (T(2*n,n)), A368046 (row sums), A368047 (alternating row sums).

Cf. A051162 (transform of n^0), A367964 (transform of n^1), this sequence (transform of n^2).

Cf. A368043, A143216.

Programs

Mathematica

Module[{n=1},NestList[Append[#+n^2,Last[#]+2(n++^2)]&,{0},10]] (* or *)
Table[(k(k+1)(2k+1)+n(n+1)(2n+1))/6,{n,0,10},{k,0,n}] (* Paolo Xausa, Dec 10 2023 *)

Python

from functools import cache
@cache
def Trow(n: int) -> list[int]:
    if n == 0: return [0]
    row = Trow(n - 1) + [0]
    for k in range(n): row[k] += n * n
    row[n] = row[n - 1] + n * n
    return row
print([k for n in range(10) for k in Trow(n)])

Formula

T(n, k) = A000330(k) + A000330(n).