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

%I A176793 #10 Oct 03 2024 06:28:16
%S A176793 1,1,1,1,5,1,1,25,25,1,1,113,145,113,1,1,481,673,673,481,1,1,1985,
%T A176793 2881,3137,2881,1985,1,1,8065,11905,13441,13441,11905,8065,1,1,32513,
%U A176793 48385,55553,57601,55553,48385,32513,1,1,130561,195073,225793,238081,238081,225793,195073,130561,1
%N A176793 A symmetrical triangle read by rows: T(n, k) = 2^n*(q^k - 1)*(q^(n - k) - 1) + 1, where q = 2.
%H A176793 G. C. Greubel, <a href="/A176793/b176793.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A176793 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.
%F A176793 From _G. C. Greubel_, Oct 02 2024: (Start)
%F A176793 T(n, k) = 2^n*(2^k - 1)*(2^(n-k) - 1) + 1.
%F A176793 T(2*n, n) = 1 + 4^n - 2*8^n + 16^n = 1 + 4*A110206(n).
%F A176793 Sum_{k=0..n} T(n, k) = 4^n*(n-3) + 2^n*(n+3) + (n+1).
%F A176793 Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*(1+(-1)^n)*(1 - (2/3)*binomial(2^n, 2)). (End)
%e A176793 Triangle begins as:
%e A176793   1;
%e A176793   1,      1;
%e A176793   1,      5,      1;
%e A176793   1,     25,     25,      1;
%e A176793   1,    113,    145,    113,      1;
%e A176793   1,    481,    673,    673,    481,      1;
%e A176793   1,   1985,   2881,   3137,   2881,   1985,      1;
%e A176793   1,   8065,  11905,  13441,  13441,  11905,   8065,      1;
%e A176793   1,  32513,  48385,  55553,  57601,  55553,  48385,  32513,      1;
%e A176793   1, 130561, 195073, 225793, 238081, 238081, 225793, 195073, 130561,    1;
%t A176793 T[n_, k_, q_] := 2^n*(q^k - 1)*(q^(n - k) - 1) + 1;
%t A176793 Table[T[n, k, 2], {n,0,12}, {k,0,n}]//Flatten
%o A176793 (Magma)
%o A176793 f:= func< n,k,q | 1 + (q^k-1)*(q^(n-k)-1)*2^n >;
%o A176793 A176793:= func< n,k | f(n,k,2) >;
%o A176793 [A176793(n,k): k in [0..n], n in [0..13]]; // _G. C. Greubel_, Oct 02 2024
%o A176793 (SageMath)
%o A176793 def f(n, k, q): return 1 + (q^k -1)*(q^(n-k) -1)*2^n
%o A176793 def A176793(n,k): return f(n,k,2)
%o A176793 flatten([[A176793(n, k) for k in range(n+1)] for n in range(14)]) # _G. C. Greubel_, Oct 02 2024
%Y A176793 Cf. A000012 (q=1), this sequence (q=2), A176794 (q=3), A176795 (q=4).
%Y A176793 Cf. A110206.
%K A176793 nonn,tabl
%O A176793 0,5
%A A176793 _Roger L. Bagula_, Apr 26 2010
%E A176793 Edited by _G. C. Greubel_, Oct 02 2024