A117438 - OEIS

%I #8 Jun 02 2021 22:17:18

%S 1,-4,1,16,-12,1,-64,80,-24,1,256,-448,240,-40,1,-1024,2304,-1792,560,

%T -60,1,4096,-11264,11520,-5376,1120,-84,1,-16384,53248,-67584,42240,

%U -13440,2016,-112,1,65536,-245760,372736,-292864,126720,-29568,3360,-144,1

%N Triangle T(n, k) = binomial(2*n-k, k)*(-4)^(n-k), read by rows.

%H G. C. Greubel, <a href="/A117438/b117438.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = binomial(2*n-k, k)*(-4)^(n-k).

%F Sum_{k=0..n} T(n, k) = (-1)^n*(2*n+1).

%F Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A117439(n) (upward diagonal sums).

%F T(n, k) = A117435(2*n-k, k).

%e Triangle begins

%e       1;

%e      -4,      1;

%e      16,    -12,     1;

%e     -64,     80,   -24,     1;

%e     256,   -448,   240,   -40,    1;

%e   -1024,   2304, -1792,   560,  -60,   1;

%e    4096, -11264, 11520, -5376, 1120, -84, 1;

%t Table[Binomial[2*n-k, k]*(-4)^(n-k), {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 01 2021 *)

%o (Sage) flatten([[binomial(2*n-k, k)*(-4)^(n-k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 01 2021

%Y Cf. A117435, A117439.

%K easy,sign,tabl

%O 0,2

%A _Paul Barry_, Mar 16 2006