A117438 Triangle T(n, k) = binomial(2*n-k, k)*(-4)^(n-k), read by rows.

1, -4, 1, 16, -12, 1, -64, 80, -24, 1, 256, -448, 240, -40, 1, -1024, 2304, -1792, 560, -60, 1, 4096, -11264, 11520, -5376, 1120, -84, 1, -16384, 53248, -67584, 42240, -13440, 2016, -112, 1, 65536, -245760, 372736, -292864, 126720, -29568, 3360, -144, 1

Offset: 0

Author

Paul Barry, Mar 16 2006

Keywords

• easy
• sign
• tabl

Examples

			Triangle begins
      1;
     -4,      1;
     16,    -12,     1;
    -64,     80,   -24,     1;
    256,   -448,   240,   -40,    1;
  -1024,   2304, -1792,   560,  -60,   1;
   4096, -11264, 11520, -5376, 1120, -84, 1;
		

Links

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

Crossrefs

Cf. A117435, A117439.

Programs

Mathematica

Table[Binomial[2*n-k, k]*(-4)^(n-k), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 01 2021 *)

Sage

flatten([[binomial(2*n-k, k)*(-4)^(n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 01 2021

Formula

T(n, k) = binomial(2*n-k, k)*(-4)^(n-k).

Sum_{k=0..n} T(n, k) = (-1)^n*(2*n+1).

Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A117439(n) (upward diagonal sums).

T(n, k) = A117435(2*n-k, k).