OFFSET
0,1
COMMENTS
Row sums are 2^(n+1): {2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, ...}.
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
FORMULA
T(n, k) = 2^k - binomial(n, k+1) + 2^(n-k) - binomial(n, n-k+1).
EXAMPLE
Triangle begins as:
2;
2, 2;
3, 2, 3;
6, 2, 2, 6;
13, 3, 0, 3, 13;
28, 7, -3, -3, 7, 28;
59, 18, -6, -14, -6, 18, 59;
122, 44, -6, -32, -32, -6, 44, 122;
249, 101, 4, -58, -80, -58, 4, 101, 249;
504, 221, 39, -90, -162, -162, -90, 39, 221, 504;
1015, 468, 130, -119, -292, -356, -292, -119, 130, 468, 1015;
MAPLE
f(n, m):= 2^m - binomial(n, m+1); seq(seq( f(n, k) + f(n, n-k), k=0..n), n=0..10); # G. C. Greubel, Dec 01 2019
MATHEMATICA
f[n_, m_]:= 2^m - Binomial[n, m+1]; T[n_, k_]:= f[n, k] + f[n, n-k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten
PROG
(PARI) T(n, k) = my(f(n, m)=2^m - binomial(n, m+1)); f(n, k) + f(n, n-k); \\ G. C. Greubel, Dec 01 2019
(Magma)
f:= func< n, m | 2^m - Binomial(n, m+1) >;
[f(n, k)+f(n, n-k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 01 2019
(Sage)
def f(n, m): return 2^m - binomial(n, m+1)
def T(n, k): return f(n, k) + f(n, n-k)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 01 2019
(GAP) Flat(List([0..10], n-> List([0..n], k-> 2^k - Binomial(n, k+1) + 2^(n-k) - Binomial(n, n-k+1) ))); # G. C. Greubel, Dec 01 2019
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Feb 17 2009
STATUS
approved