OFFSET
0,5
COMMENTS
Row sums are: A155559(n) = {0, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...}.
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
FORMULA
T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = binomial(n, k)*(1 - (-1)^k)/2.
From G. C. Greubel, Dec 01 2019: (Start)
T(n, k) = binomial(n, k)*(2 - (-1)^k*(1 + (-1)^n))/2.
Sum_{k=0..n} T(n,k) = 2^n = A155559(n) for n >= 1.
Sum_{k=0..n-1} T(n,k) = (2^(n+1) - (1-(-1)^n))/2 = A051049(n), n >= 1. (End)
EXAMPLE
Triangle begins as:
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, 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, 20, 0, 240, 0, 504, 0, 240, 0, 20, 0;
MAPLE
seq(seq( binomial(n, k)*(2 - (-1)^k*(1+(-1)^n))/2, k=0..n), n=0..12); # G. C. Greubel, Dec 01 2019
MATHEMATICA
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
Table[Binomial[n, k]*(2-(-1)^k*(1+(-1)^n))/2, {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 01 2019 *)
PROG
(PARI) T(n, k) = binomial(n, k)*(2 - (-1)^k*(1+(-1)^n))/2; \\ G. C. Greubel, Dec 01 2019
(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
(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
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 01 2009
STATUS
approved