A172176 Triangle T(n, k) = 1 + (n + k - n*k)*(2*n - k - n*(n-k)), read by rows.

1, 2, 2, 1, 2, 1, -8, 0, 0, -8, -31, -4, 5, -4, -31, -74, -10, 22, 22, -10, -74, -143, -18, 57, 82, 57, -18, -143, -244, -28, 116, 188, 188, 116, -28, -244, -383, -40, 205, 352, 401, 352, 205, -40, -383, -566, -54, 330, 586, 714, 714, 586, 330, -54, -566

Offset: 0

Author

Roger L. Bagula, Jan 28 2010

Keywords

• sign
• tabl
• easy

Examples

			Triangle begins as:
     1;
     2,   2;
     1,   2,   1;
    -8,   0,   0,  -8;
   -31,  -4,   5,  -4,  -31;
   -74, -10,  22,  22,  -10,  -74;
  -143, -18,  57,  82,   57,  -18, -143;
  -244, -28, 116, 188,  188,  116,  -28, -244;
  -383, -40, 205, 352,  401,  352,  205,  -40, -383;
  -566, -54, 330, 586,  714,  714,  586,  330,  -54, -566;
  -799, -70, 497, 902, 1145, 1226, 1145,  902,  497,  -70, -799;
		

Links

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

Crossrefs

Cf. A027620, A028552, A033445.

Programs

Magma

[1 + (n-(n-1)*k)*(n-(n-1)*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 26 2022

Maple

A172176:= proc(n,m) 1+(n+m-n*m)*(2*n-m-n*(n-m)); end proc:
seq(seq(A172176(n,m), m=0..n), n=0..12);

Mathematica

T[n_, k_]= 1 + (n-(n-1)*k)*(n-(n-1)*(n-k));
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten

SageMath

def A172176(n,k): return 1 + (n-(n-1)*k)*(n-(n-1)*(n-k))
flatten([[A172176(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 26 2022

Formula

T(n, k) = 1 + (n-(n-1)*k)*(n-(n-1)*(n-k)).

T(n, n-k) = T(n, k).

T(n, 0) = 1 - A027620(n-3).

T(n, 1) = -A028552(n-3).

T(n, 2) = A033445(n-2).

Sum_{k=0..n} T(n, k) = (n+1)*(n^4 - 9*n^3 + 15*n^2 - n + 6)/6.