A360665 Square array T(n, k) = k*((2*n-1)*k+1)/2 read by rising antidiagonals.

0, 0, 0, 0, 1, -1, 0, 2, 3, -3, 0, 3, 7, 6, -6, 0, 4, 11, 15, 10, -10, 0, 5, 15, 24, 26, 15, -15, 0, 6, 19, 33, 42, 40, 21, -21, 0, 7, 23, 42, 58, 65, 57, 28, -28, 0, 8, 27, 51, 74, 90, 93, 77, 36, -36, 0, 9, 31, 60, 90, 115, 129, 126, 100, 45, -45

Offset: 0

Author

Paul Curtz, Mar 17 2023

Keywords

• sign
• tabl
• easy

Examples

			By rows:
   0,   0,  -1,  -3,  -6,  -10,  -15,  -21,  -28, ...   = -A161680
   0,   1,   3,   6,  10,   15,   21,   28,   36, ...   =  A000217
   0,   2,   7,  15,  26,   40,   57,   77,  100, ...   =  A005449
   0,   3,  11,  24,  42,   65,   93,  126,  164, ...   =  A005475
   0,   4,  15,  33,  58,   90,  129,  175,  228, ...   =  A022265
   0,   5,  19,  42,  74,  115,  165,  224,  292, ...   =  A022267
   0,   6,  23,  51,  90,  140,  201,  273,  356, ...   =  A022269
   ... .
		

Crossrefs

Cf. Antidiagonal sums: A034827(n+1).

Cf. A000217, A000290, A005449, A005475, A022265.

Cf. A022267, A022269, A059270, A081436, A101986.

Cf. A161680, A162254, A179297, A360962, A361226.

Programs

Mathematica

T[n_, k_] := ((2*n - 1)*k^2 + k)/2; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Mar 31 2023 *)

PARI

T(n, k) = ((2*n-1)*k^2+k)/2 \\ Thomas Scheuerle, Mar 17 2023

Formula

T(n,k) = T(n,k-1)+k^2.

T(n,n) = A081436(n-1).

T(n,n+1) = A059270(n).

T(n,n+4) = -3*A179297(n+4).

T(n+3,n) = A162254(n).

T(n+5,n) = 3*A101986(n).

From Stefano Spezia, Mar 31 2023: (Start)

O.g.f.: (x*y - y^2 + 2*x*y^2)/((1 - x)^2*(1 - y)^3).

E.g.f.: exp(x+y)*y*(2*x - y + 2*x*y)/2. (End)