A166555 - OEIS

%I #16 Dec 04 2024 13:53:02

%S 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,

%T 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,

%U 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

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

%C Number of positive terms in n-th row (n>=0) equals to A000120(n). - _Vladimir Shevelev_, Oct 25 2010

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

%H G. C. Greubel, <a href="/A166555/b166555.txt">Rows n = 0..100 of the triangle, flattened</a>

%F 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.

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

%F From _G. C. Greubel_, Dec 02 2024: (Start)

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

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

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

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

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

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

%e First few rows of the triangle are:

%e   1;

%e   1, 2;

%e   1, 0, 4;

%e   1, 2, 4, 8;

%e   1, 0, 0, 0, 16;

%e   1, 2, 0, 0, 16,.32;

%e   1, 0, 4, 0, 16,..0,..64;

%e   1, 2, 4, 8, 16,.32,..64,..128;

%e   1, 0, 0, 0,..0,..0,...0,....0,..256;

%e   1, 2, 0, 0,..0,..0,...0,....0,..256,...512;

%e   1, 0, 4, 0,..0,..0,...0,....0,..256,.....0,...1024;

%e   1, 2, 4, 8,..0,..0,...0,....0,..256,...512,...1024,...2048;

%e   1, 0, 0, 0, 16,..0,...0,....0,..256,.....0,......0,......0,..4096;

%e   ...

%t A166555[n_, k_]:= 2^k*Mod[Binomial[n, k], 2];

%t Table[A166555[n,k], {n,0,14}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 01 2024 *)

%o (Magma)

%o A166555:= func< n,k | 2^k*( Binomial(n,k) mod 2) >;

%o [A166555(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Dec 01 2024

%o (SageMath)

%o def A166555(n,k): return 2^k*int(not ~n & k) if k<n+1 else 0

%o print(flatten([[A166555(n,k) for k in range(n+1)] for n in range(15)])) # _G. C. Greubel_, Dec 01 2024

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

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

%K nonn,tabl

%O 0,3

%A _Gary W. Adamson_, Oct 17 2009

%E New name by _G. C. Greubel_, Dec 02 2024