A152439 Irregular triangle T(n, k) = (n-3)*(k-n)*(k-n-2) - (2*n-k)*(k-2), with 0 <= k <= 2*n.

0, -2, 1, 2, 0, 0, 0, 0, 0, 12, 5, 0, -3, -4, -3, 0, 40, 22, 8, -2, -8, -10, -8, -2, 8, 90, 57, 30, 9, -6, -15, -18, -15, -6, 9, 30, 168, 116, 72, 36, 8, -12, -24, -28, -24, -12, 8, 36, 72, 280, 205, 140, 85, 40, 5, -20, -35, -40, -35, -20, 5, 40, 85, 140

Offset

0,2

Comments

Row sums are: {0, 1, 0, 7, 48, 165, 416, 875, 1632, 2793, 4480, ...}.

Links

G. C. Greubel, Rows n = 0..50 of triangle, flattened

Formula

T(n, k) = 4*( (2*n-3)*k*(k-1) - n*(n-1) + k*(k-1)) = 4*( (2*n-3)*k*(k-1) - (n-k)*(n+k-1) ) with n and k ranging over half-integer steps.

T(n, k) = (n-3)*(k-n)*(k-n-2) - (2*n-k)*(k-2), with 0 <= k <= 2*n, n >= 0. - G. C. Greubel, Dec 03 2019

Example

Irregular triangle begins as:
    0;
   -2,   1,   2;
    0,   0,   0,  0,  0;
   12,   5,   0, -3, -4,  -3,   0;
   40,  22,   8, -2, -8, -10,  -8,  -2,   8;
   90,  57,  30,  9, -6, -15, -18, -15,  -6,   9,  30;
  168, 116,  72, 36,  8, -12, -24, -28, -24, -12,   8, 36, 72;
  280, 205, 140, 85, 40,   5, -20, -35, -40, -35, -20,  5, 40, 85, 140;

Maple

seq(seq( (n-3)*(k-n)*(k-n-2) -(2*n-k)*(k-2), k=0..2*n), n=0..10); # G. C. Greubel, Dec 03 2019

Mathematica

T[n_, k_]:= 4*((2*n-3)*k*(k-1) - (n-k)*(n+k-1)); Table[T[n, k], {n, 0, 5, 1/2}, {k, -n, n, 1/2}]//Flatten
T[n_, k_]:= (n-3)*(k-n)*(k-n-2) -(2*n-k)*(k-2); Table[T[n, k], {n, 0, 10}, {k, 0, 2*n}]//Flatten (* G. C. Greubel, Dec 03 2019 *)

Prog

(PARI) T(n, k) = (n-3)*(k-n)*(k-n-2) -(2*n-k)*(k-2); \\ G. C. Greubel, Dec 03 2019
(Magma) [(n-3)*(k-n)*(k-n-2) -(2*n-k)*(k-2): k in [0..2*n], n in [0..10]]; // G. C. Greubel, Dec 03 2019
(Sage) [[(n-3)*(k-n)*(k-n-2) -(2*n-k)*(k-2) for k in (0..2*n)] for n in (0..10)] # G. C. Greubel, Dec 03 2019
(GAP) Flat(List([0..10], n-> List([0..2*n], k-> (n-3)*(k-n)*(k-n-2) -(2*n-k)*(k-2) ))); # G. C. Greubel, Dec 03 2019

Crossrefs

Cf. A152420.

Sequence in context: A164810 A322392 A089538 * A070965 A079548 A175620

Adjacent sequences: A152436 A152437 A152438 * A152440 A152441 A152442

Keyword

tabf,sign

Author

Roger L. Bagula, Dec 04 2008

Extensions

Edited by G. C. Greubel, Dec 03 2019

Status

approved