A166555 Triangle read by rows, T(n, k) = 2^k * A047999(n, k).

1, 1, 2, 1, 0, 4, 1, 2, 4, 8, 1, 0, 0, 0, 16, 1, 2, 0, 0, 16, 32, 1, 0, 4, 0, 16, 0, 64, 1, 2, 4, 8, 16, 32, 64, 128, 1, 0, 0, 0, 0, 0, 0, 0, 256, 1, 2, 0, 0, 0, 0, 0, 0, 256, 512, 1, 0, 4, 0, 0, 0, 0, 0, 256, 0, 1024, 1, 2, 4, 8, 0, 0, 0, 0, 256, 512, 1024, 2048, 1, 0, 0, 0, 16, 0, 0, 0, 256, 0, 0, 0, 4096

Offset: 0

Author

Gary W. Adamson, Oct 17 2009

Keywords

• nonn
• tabl

Comments

Number of positive terms in n-th row (n>=0) equals to A000120(n). - Vladimir Shevelev, Oct 25 2010

Former name: Triangle read by rows, Sierpinski's gasket, A047999 * (1,2,4,8,...) diagonalized. - G. C. Greubel, Dec 02 2024

Examples

			First few rows of the triangle are:
  1;
  1, 2;
  1, 0, 4;
  1, 2, 4, 8;
  1, 0, 0, 0, 16;
  1, 2, 0, 0, 16,.32;
  1, 0, 4, 0, 16,..0,..64;
  1, 2, 4, 8, 16,.32,..64,..128;
  1, 0, 0, 0,..0,..0,...0,....0,..256;
  1, 2, 0, 0,..0,..0,...0,....0,..256,...512;
  1, 0, 4, 0,..0,..0,...0,....0,..256,.....0,...1024;
  1, 2, 4, 8,..0,..0,...0,....0,..256,...512,...1024,...2048;
  1, 0, 0, 0, 16,..0,...0,....0,..256,.....0,......0,......0,..4096;
  ...
		

Links

• G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Crossrefs

Cf. A000007, A000079, A000120, A038183, A147999.

Sums include: A001317 (row), A101624 (diagonal), A101625 (odd rows of signed diagonal).

Programs

Magma

A166555:= func< n,k | 2^k*( Binomial(n,k) mod 2) >;
[A166555(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 01 2024

Mathematica

A166555[n_, k_]:= 2^k*Mod[Binomial[n, k], 2];
Table[A166555[n,k], {n,0,14}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 01 2024 *)

SageMath

def A166555(n,k): return 2^k*int(not ~n & k) if kA166555(n,k) for k in range(n+1)] for n in range(15)])) # G. C. Greubel, Dec 01 2024

Formula

Triangle read by rows, A047999 * Q. A047999 = Sierpinski's gasket, Q = an infinite lower triangular matrix with (1,2,4,8,...) as the main diagonal and the rest zeros.

Sum_{k=0..n} T(n, k) = A001317(n).

From G. C. Greubel, Dec 02 2024: (Start)

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

T(n, n) = A000079(n).

T(2*n, n) = A000007(n).

Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*( (1+(-1)^n)*A038183(n/2) - (1-(-1)^n) *A038183((n-1)/2) ).

Sum_{k=0..floor(n/2)} T(n-k, k) = A101624(n).

Sum_{k=0..floor((2*m+1)/2)} T(2*m-k+1, k) = A101625(m+1), m >= 0. (End)

Extensions

New name by G. C. Greubel, Dec 02 2024