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A155997 Triangle read by rows: T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = binomial(n, k)*(1 + (-1)^k)/2.

%I A155997 #8 Sep 08 2022 08:45:41
%S A155997 2,1,1,2,0,2,1,3,3,1,2,0,12,0,2,1,5,10,10,5,1,2,0,30,0,30,0,2,1,7,21,
%T A155997 35,35,21,7,1,2,0,56,0,140,0,56,0,2,1,9,36,84,126,126,84,36,9,1,2,0,
%U A155997 90,0,420,0,420,0,90,0,2,1,11,55,165,330,462,462,330,165,55,11,1
%N A155997 Triangle read by rows: T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = binomial(n, k)*(1 + (-1)^k)/2.
%C A155997 Row sums are: {2, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,...}
%H A155997 G. C. Greubel, <a href="/A155997/b155997.txt">Rows n = 0..100 of triangle, flattened</a>
%F A155997 T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = binomial(n, k)*(1 + (-1)^k)/2.
%F A155997 From _G. C. Greubel_, Dec 01 2019: (Start)
%F A155997 T(n, k) = binomial(n, k)*(2 + (-1)^k*(1 + (-1)^n))/2.
%F A155997 Sum_{k=0..n} T(n,k) = 2^n for n >= 1.
%F A155997 Sum_{k=0..n-1} T(n,k) = (2^(n+1) - 3 - (-1)^n)/2 = A140253(n), n >= 2. (End)
%e A155997 Triangle begins as:
%e A155997   2;
%e A155997   1, 1;
%e A155997   2, 0,  2;
%e A155997   1, 3,  3,  1;
%e A155997   2, 0, 12,  0,   2;
%e A155997   1, 5, 10, 10,   5,   1;
%e A155997   2, 0, 30,  0,  30,   0,   2;
%e A155997   1, 7, 21, 35,  35,  21,   7,  1;
%e A155997   2, 0, 56,  0, 140,   0,  56,  0,  2;
%e A155997   1, 9, 36, 84, 126, 126,  84, 36,  9, 1;
%e A155997   2, 0, 90,  0, 420,   0, 420,  0, 90, 0, 2;
%p A155997 seq(seq( binomial(n, k)*(2+(-1)^k*(1+(-1)^n))/2, k=0..n), n=0..12); # _G. C. Greubel_, Dec 01 2019
%t A155997 f[n_, k_]:= (Binomial[n, k] + (-1)^k*Binomial[n, k])/2; Table[f[n,k]+f[n,n-k], {n,0,10}, {k,0,n}]//Flatten
%t A155997 Table[Binomial[n, k]*(2+(-1)^k*(1+(-1)^n))/2, {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 01 2019 *)
%o A155997 (PARI) T(n,k) = binomial(n, k)*(2 + (-1)^k*(1 + (-1)^n))/2; \\ _G. C. Greubel_, Dec 01 2019
%o A155997 (Magma) [Binomial(n, k)*(2+(-1)^k*(1+(-1)^n))/2: k in [0..n], n in [0..12]]; // _G. C. Greubel_, Dec 01 2019
%o A155997 (Sage) [[binomial(n, k)*(2+(-1)^k*(1+(-1)^n))/2 for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Dec 01 2019
%o A155997 (GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n, k)*(2 + (-1)^k*(1 + (-1)^n))/2 ))); # _G. C. Greubel_, Dec 01 2019
%K A155997 nonn,tabl
%O A155997 0,1
%A A155997 _Roger L. Bagula_, Feb 01 2009