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A155998 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 A155998 #6 Sep 08 2022 08:45:41
%S A155998 0,1,1,0,4,0,1,3,3,1,0,8,0,8,0,1,5,10,10,5,1,0,12,0,40,0,12,0,1,7,21,
%T A155998 35,35,21,7,1,0,16,0,112,0,112,0,16,0,1,9,36,84,126,126,84,36,9,1,0,
%U A155998 20,0,240,0,504,0,240,0,20,0
%N A155998 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 A155998 Row sums are: A155559(n) = {0, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...}.
%H A155998 G. C. Greubel, <a href="/A155998/b155998.txt">Rows n = 0..100 of triangle, flattened</a>
%F A155998 T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = binomial(n, k)*(1 - (-1)^k)/2.
%F A155998 From _G. C. Greubel_, Dec 01 2019: (Start)
%F A155998 T(n, k) = binomial(n, k)*(2 - (-1)^k*(1 + (-1)^n))/2.
%F A155998 Sum_{k=0..n} T(n,k) = 2^n = A155559(n) for n >= 1.
%F A155998 Sum_{k=0..n-1} T(n,k) = (2^(n+1) - (1-(-1)^n))/2 = A051049(n), n >= 1. (End)
%e A155998 Triangle begins as:
%e A155998   0;
%e A155998   1,  1;
%e A155998   0,  4,  0;
%e A155998   1,  3,  3,   1;
%e A155998   0,  8,  0,   8,   0;
%e A155998   1,  5, 10,  10,   5,   1;
%e A155998   0, 12,  0,  40,   0,  12,  0;
%e A155998   1,  7, 21,  35,  35,  21,  7,   1;
%e A155998   0, 16,  0, 112,   0, 112,  0,  16, 0;
%e A155998   1,  9, 36,  84, 126, 126, 84,  36, 9,  1;
%e A155998   0, 20,  0, 240,   0, 504,  0, 240, 0, 20, 0;
%p A155998 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 A155998 f[n_, k_]:= Binomial[n, k]*(1 - (-1)^k)/2; Table[f[n,k]+f[n,n-k], {n, 0, 10}, {k, 0, n}]//Flatten
%t A155998 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 A155998 (PARI) T(n,k) = binomial(n, k)*(2 - (-1)^k*(1+(-1)^n))/2; \\ _G. C. Greubel_, Dec 01 2019
%o A155998 (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 A155998 (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 A155998 (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 A155998 nonn,tabl
%O A155998 0,5
%A A155998 _Roger L. Bagula_, Feb 01 2009