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%I #8 Sep 08 2022 08:45:41
%S 0,-1,1,-2,-2,4,-3,-6,-2,11,-4,-11,-12,1,26,-5,-17,-27,-19,11,57,-6,
%T -24,-48,-54,-24,36,120,-7,-32,-76,-110,-94,-20,92,247,-8,-41,-112,
%U -194,-220,-146,8,211,502,-9,-51,-157,-314,-430,-398,-202,91,457,1013
%N Triangle read by rows: T(n, k) = 2^k - binomial(n+1, k+1) - ((2*k-n)/(k+1)) * binomial(n+1, k).
%C Row sums are zero.
%H G. C. Greubel, <a href="/A156864/b156864.txt">Rows n = 1..100 of triangle, flattened</a>
%F T(n, k) = 2^k - binomial(n+1, k+1) - ((2*k-n)/(k+1)) * binomial(n+1, k).
%F From _G. C. Greubel_, Dec 01 2019: (Start)
%F T(n, n) = 2^n - n - 1 = A000295(n).
%F Sum_{k=1..n-1} T(n,k) = - A000295(n). (End)
%e Triangle begins as:
%e 0;
%e -1, 1;
%e -2, -2, 4;
%e -3, -6, -2, 11;
%e -4, -11, -12, 1, 26;
%e -5, -17, -27, -19, 11, 57;
%e -6, -24, -48, -54, -24, 36, 120;
%e -7, -32, -76, -110, -94, -20, 92, 247;
%e -8, -41, -112, -194, -220, -146, 8, 211, 502;
%e -9, -51, -157, -314, -430, -398, -202, 91, 457, 1013;
%e -10, -62, -212, -479, -760, -860, -664, -239, 292, 958, 2036;
%p 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
%t 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
%o (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
%o (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
%o (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
%o (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
%K sign,tabl
%O 1,4
%A _Roger L. Bagula_, Feb 17 2009
%E Edited by _G. C. Greubel_, Dec 01 2019
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