A173787 Triangle read by rows: T(n,k) = 2^n - 2^k, 0 <= k <= n.

0, 1, 0, 3, 2, 0, 7, 6, 4, 0, 15, 14, 12, 8, 0, 31, 30, 28, 24, 16, 0, 63, 62, 60, 56, 48, 32, 0, 127, 126, 124, 120, 112, 96, 64, 0, 255, 254, 252, 248, 240, 224, 192, 128, 0, 511, 510, 508, 504, 496, 480, 448, 384, 256, 0, 1023, 1022, 1020, 1016, 1008, 992, 960, 896, 768, 512, 0

Offset: 0

Author

Reinhard Zumkeller, Feb 28 2010

Keywords

• nonn
• easy
• tabl

Examples

			Triangle begins as:
   0;
   1,  0;
   3,  2,  0;
   7,  6,  4,  0;
  15, 14, 12,  8,  0;
  31, 30, 28, 24, 16, 0;
		

Links

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

Programs

Magma

[2^n -2^k: k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 13 2021

Mathematica

Table[2^n -2^k, {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 13 2021 *)

Sage

flatten([[2^n -2^k for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 13 2021

Formula

A000120(T(n,k)) = A025581(n,k).

Row sums give A000337.

Central terms give A020522.

T(2*n+1, n) = A006516(n+1).

T(2*n+3, n+2) = A059153(n).

T(n, k) = A140513(n,k) - A173786(n,k), 0 <= k <= n.

T(n, k) = A173786(n,k) - A059268(n+1,k+1), 0 < k <= n.

T(2*n, 2*k) = T(n,k) * A173786(n,k), 0 <= k <= n.

T(n, 0) = A000225(n).

T(n, 1) = A000918(n) for n>0.

T(n, 2) = A028399(n) for n>1.

T(n, 3) = A159741(n-3) for n>3.

T(n, 4) = A175164(n-4) for n>4.

T(n, 5) = A175165(n-5) for n>5.

T(n, 6) = A175166(n-6) for n>6.

T(n, n-4) = A110286(n-4) for n>3.

T(n, n-3) = A005009(n-3) for n>2.

T(n, n-2) = A007283(n-2) for n>1.

T(n, n-1) = A000079(n-1) for n>0.

T(n, n) = A000004(n).