A176795 Triangle read by rows: T(n, k) = 2^n*(q^k - 1)*(q^(n - k) - 1) + 1, where q = 4.

1, 1, 1, 1, 37, 1, 1, 361, 361, 1, 1, 3025, 3601, 3025, 1, 1, 24481, 30241, 30241, 24481, 1, 1, 196417, 244801, 254017, 244801, 196417, 1, 1, 1572481, 1964161, 2056321, 2056321, 1964161, 1572481, 1, 1, 12582145, 15724801, 16498945, 16646401, 16498945, 15724801, 12582145, 1

Offset: 0

Author

Roger L. Bagula, Apr 26 2010

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,        1;
  1,       37,        1;
  1,      361,      361,        1;
  1,     3025,     3601,     3025,        1;
  1,    24481,    30241,    30241,    24481,        1;
  1,   196417,   244801,   254017,   244801,   196417,        1;
  1,  1572481,  1964161,  2056321,  2056321,  1964161,  1572481,        1;
  1, 12582145, 15724801, 16498945, 16646401, 16498945, 15724801, 12582145, 1;
		

Links

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

Crossrefs

Cf. A000012 (q=1), A176793 (q=2), A176794 (q=3), this sequence (q=4).

Programs

Magma

f:= func< n,k,q | 1 + (q^k-1)*(q^(n-k)-1)*2^n >;
A176795:= func< n,k | f(n,k,4) >;
[A176795(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Oct 03 2024

Mathematica

T[n_, k_, q_] := 2^n*(q^k - 1)*(q^(n - k) - 1) + 1;Table[T[n, k, 4], {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 A176795(n,k): return f(n,k,4)
flatten([[A176795(n, k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Oct 03 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 = 4.

From G. C. Greubel, Oct 03 2024: (Start)

T(n, k) = 2^n*(4^k - 1)*(4^(n-k) - 1) + 1.

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

Sum_{k=0..n} T(n, k) = (1/3)*((3*n + 5)*2^n + (3*n - 5)*8^n) + (n + 1).

Sum_{k=0..n} (-1)^k*T(n, k) = (1/10)*(1 + (-1)^n)*(5 + 3*2^n - 3*8^n). (End)

Extensions

Edited by G. C. Greubel, Oct 04 2024