login
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.
1

%I #6 Sep 08 2022 08:45:41

%S 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 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 20,0,240,0,504,0,240,0,20,0

%N 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 Row sums are: A155559(n) = {0, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...}.

%H G. C. Greubel, <a href="/A155998/b155998.txt">Rows n = 0..100 of triangle, flattened</a>

%F T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = binomial(n, k)*(1 - (-1)^k)/2.

%F From _G. C. Greubel_, Dec 01 2019: (Start)

%F T(n, k) = binomial(n, k)*(2 - (-1)^k*(1 + (-1)^n))/2.

%F Sum_{k=0..n} T(n,k) = 2^n = A155559(n) for n >= 1.

%F Sum_{k=0..n-1} T(n,k) = (2^(n+1) - (1-(-1)^n))/2 = A051049(n), n >= 1. (End)

%e Triangle begins as:

%e 0;

%e 1, 1;

%e 0, 4, 0;

%e 1, 3, 3, 1;

%e 0, 8, 0, 8, 0;

%e 1, 5, 10, 10, 5, 1;

%e 0, 12, 0, 40, 0, 12, 0;

%e 1, 7, 21, 35, 35, 21, 7, 1;

%e 0, 16, 0, 112, 0, 112, 0, 16, 0;

%e 1, 9, 36, 84, 126, 126, 84, 36, 9, 1;

%e 0, 20, 0, 240, 0, 504, 0, 240, 0, 20, 0;

%p 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 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 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 (PARI) T(n,k) = binomial(n, k)*(2 - (-1)^k*(1+(-1)^n))/2; \\ _G. C. Greubel_, Dec 01 2019

%o (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 (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 (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 nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 01 2009