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

%I A176795 #8 Oct 04 2024 07:41:24
%S A176795 1,1,1,1,37,1,1,361,361,1,1,3025,3601,3025,1,1,24481,30241,30241,
%T A176795 24481,1,1,196417,244801,254017,244801,196417,1,1,1572481,1964161,
%U A176795 2056321,2056321,1964161,1572481,1,1,12582145,15724801,16498945,16646401,16498945,15724801,12582145,1
%N A176795 Triangle read by rows: T(n, k) = 2^n*(q^k - 1)*(q^(n - k) - 1) + 1, where q = 4.
%H A176795 G. C. Greubel, <a href="/A176795/b176795.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A176795 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.
%F A176795 From _G. C. Greubel_, Oct 03 2024: (Start)
%F A176795 T(n, k) = 2^n*(4^k - 1)*(4^(n-k) - 1) + 1.
%F A176795 T(2*n, n) = 1 + 4^n*(4^n - 1)^2.
%F A176795 Sum_{k=0..n} T(n, k) = (1/3)*((3*n + 5)*2^n + (3*n - 5)*8^n) + (n + 1).
%F A176795 Sum_{k=0..n} (-1)^k*T(n, k) = (1/10)*(1 + (-1)^n)*(5 + 3*2^n - 3*8^n). (End)
%e A176795 Triangle begins as:
%e A176795   1;
%e A176795   1,        1;
%e A176795   1,       37,        1;
%e A176795   1,      361,      361,        1;
%e A176795   1,     3025,     3601,     3025,        1;
%e A176795   1,    24481,    30241,    30241,    24481,        1;
%e A176795   1,   196417,   244801,   254017,   244801,   196417,        1;
%e A176795   1,  1572481,  1964161,  2056321,  2056321,  1964161,  1572481,        1;
%e A176795   1, 12582145, 15724801, 16498945, 16646401, 16498945, 15724801, 12582145, 1;
%t A176795 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
%o A176795 (Magma)
%o A176795 f:= func< n,k,q | 1 + (q^k-1)*(q^(n-k)-1)*2^n >;
%o A176795 A176795:= func< n,k | f(n,k,4) >;
%o A176795 [A176795(n,k): k in [0..n], n in [0..13]]; // _G. C. Greubel_, Oct 03 2024
%o A176795 (SageMath)
%o A176795 def f(n, k, q): return 1 + (q^k -1)*(q^(n-k) -1)*2^n
%o A176795 def A176795(n,k): return f(n,k,4)
%o A176795 flatten([[A176795(n, k) for k in range(n+1)] for n in range(14)]) # _G. C. Greubel_, Oct 03 2024
%Y A176795 Cf. A000012 (q=1), A176793 (q=2), A176794 (q=3), this sequence (q=4).
%K A176795 nonn,tabl
%O A176795 0,5
%A A176795 _Roger L. Bagula_, Apr 26 2010
%E A176795 Edited by _G. C. Greubel_, Oct 04 2024