OFFSET
1,4
COMMENTS
Row sums are zero.
LINKS
G. C. Greubel, Rows n = 1..100 of triangle, flattened
FORMULA
T(n, k) = 2^k - binomial(n+1, k+1) - ((2*k-n)/(k+1)) * binomial(n+1, k).
From G. C. Greubel, Dec 01 2019: (Start)
T(n, n) = 2^n - n - 1 = A000295(n).
Sum_{k=1..n-1} T(n,k) = - A000295(n). (End)
EXAMPLE
Triangle begins as:
0;
-1, 1;
-2, -2, 4;
-3, -6, -2, 11;
-4, -11, -12, 1, 26;
-5, -17, -27, -19, 11, 57;
-6, -24, -48, -54, -24, 36, 120;
-7, -32, -76, -110, -94, -20, 92, 247;
-8, -41, -112, -194, -220, -146, 8, 211, 502;
-9, -51, -157, -314, -430, -398, -202, 91, 457, 1013;
-10, -62, -212, -479, -760, -860, -664, -239, 292, 958, 2036;
MAPLE
b:=binomial; seq(seq( 2^k -b(n+1, k+1) -((2*k-n)/(k+1))*b(n+1, k), k=1..n), n=1..12); # G. C. Greubel, Dec 01 2019
MATHEMATICA
T[n_, k_]:= 2^k - Binomial[n+1, k+1] - ((2*k-n)/(k+1))*Binomial[n+1, k]; Table[T[n, k], {n, 12}, {k, n}]//Flatten
PROG
(PARI) T(n, k) = 2^k -binomial(n+1, k+1) -((2*k-n)/(k+1))*binomial(n+1, k); \\ G. C. Greubel, Dec 01 2019
(Magma) B:=Binomial; [2^k -B(n+1, k+1) -((2*k-n)/(k+1))*B(n+1, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Dec 01 2019
(Sage) b=binomial; [[2^k -b(n+1, k+1) -((2*k-n)/(k+1))*b(n+1, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Dec 01 2019
(GAP) B:=Binomial;; Flat(List([1..12], n-> List([1..n], k-> 2^k - B(n+1, k+1) - ((2*k-n)/(k+1))*B(n+1, k) ))); # G. C. Greubel, Dec 01 2019
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Feb 17 2009
EXTENSIONS
Edited by G. C. Greubel, Dec 01 2019
STATUS
approved