A176793 A symmetrical triangle read by rows: T(n, k) = 2^n*(q^k - 1)*(q^(n - k) - 1) + 1, where q = 2.

1, 1, 1, 1, 5, 1, 1, 25, 25, 1, 1, 113, 145, 113, 1, 1, 481, 673, 673, 481, 1, 1, 1985, 2881, 3137, 2881, 1985, 1, 1, 8065, 11905, 13441, 13441, 11905, 8065, 1, 1, 32513, 48385, 55553, 57601, 55553, 48385, 32513, 1, 1, 130561, 195073, 225793, 238081, 238081, 225793, 195073, 130561, 1

Offset: 0

Author

Roger L. Bagula, Apr 26 2010

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,      1;
  1,      5,      1;
  1,     25,     25,      1;
  1,    113,    145,    113,      1;
  1,    481,    673,    673,    481,      1;
  1,   1985,   2881,   3137,   2881,   1985,      1;
  1,   8065,  11905,  13441,  13441,  11905,   8065,      1;
  1,  32513,  48385,  55553,  57601,  55553,  48385,  32513,      1;
  1, 130561, 195073, 225793, 238081, 238081, 225793, 195073, 130561,    1;
		

Links

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

Crossrefs

Cf. A000012 (q=1), this sequence (q=2), A176794 (q=3), A176795 (q=4).

Cf. A110206.

Programs

Magma

f:= func< n,k,q | 1 + (q^k-1)*(q^(n-k)-1)*2^n >;
A176793:= func< n,k | f(n,k,2) >;
[A176793(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Oct 02 2024

Mathematica

T[n_, k_, q_] := 2^n*(q^k - 1)*(q^(n - k) - 1) + 1;
Table[T[n, k, 2], {n,0,12}, {k,0,n}]//Flatten

SageMath

def f(n, k, q): return 1 + (q^k -1)*(q^(n-k) -1)*2^n
def A176793(n,k): return f(n,k,2)
flatten([[A176793(n, k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Oct 02 2024

Formula

T(n, k) = 1 - (f(n+1, 2*k+1, q) - f(n+1, 1, q)) - (f(n+1, 2*n-2*k+1, q) - f(n+1, 2*n+1, q)), where f(n, k, q) = 2^(n-1) * q^((k-1)/2), and q = 2.

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

T(n, k) = 2^n*(2^k - 1)*(2^(n-k) - 1) + 1.

T(2*n, n) = 1 + 4^n - 2*8^n + 16^n = 1 + 4*A110206(n).

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

Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*(1+(-1)^n)*(1 - (2/3)*binomial(2^n, 2)). (End)

Extensions

Edited by G. C. Greubel, Oct 02 2024