OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
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)
EXAMPLE
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;
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
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Apr 26 2010
EXTENSIONS
Edited by G. C. Greubel, Oct 04 2024
STATUS
approved