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%I #21 Mar 04 2023 17:07:25
%S 0,1,0,-1,2,0,3,-2,4,0,-5,6,-4,8,0,11,-10,12,-8,16,0,-21,22,-20,24,
%T -16,32,0,43,-42,44,-40,48,-32,64,0,-85,86,-84,88,-80,96,-64,128,0,
%U 171,-170,172,-168,176,-160,192,-128,256,0,-341,342,-340,344,-336,352,-320,384,-256,512,0
%N Triangle T(n,k) read by rows, the k-th term of the n-th differences of the Jacobsthal sequence A001045.
%C A variant of the triangle A140503, now including the diagonal.
%C Since the diagonal contains zeros, rows sums are those of A140503.
%H G. C. Greubel, <a href="/A140944/b140944.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = T(n-1, k+1) - T(n-1, k). T(0, k) = A001045(k).
%F T(n, k) = (2^k - 2^n*(-1)^(n+k))/3, for n >= k >= 0. - _Jianing Song_, Aug 11 2022
%F From _G. C. Greubel_, Feb 18 2023: (Start)
%F T(n, n-1) = A000079(n).
%F T(2*n, n) = (-1^(n+1)*A192382(n+1).
%F T(2*n, n-1) = (-1)^n*A246036(n-1).
%F T(2*n, n+1) = A083086(n).
%F T(3*n, n) = -A115489(n).
%F Sum_{k=0..n} T(n, k) = A052992(n)*[n>0] + 0*[n=0].
%F Sum_{k=0..n} (-1)^k*T(n, k) = A045883(n).
%F Sum_{k=0..n} 2^k*T(n, k) = A084175(n).
%F Sum_{k=0..n} (-2)^k*T(n, k) = (-1)^(n+1)*A109765(n).
%F Sum_{k=0..n} 3^k*T(n, k) = A091056(n+1).
%F Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^(n+1)*A097038(n).
%F Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^(n+1)*A138495(n). (End)
%e Triangle begins as:
%e 0;
%e 1, 0;
%e -1, 2, 0;
%e 3, -2, 4, 0;
%e -5, 6, -4, 8, 0;
%e 11, -10, 12, -8, 16, 0;
%e -21, 22, -20, 24, -16, 32, 0;
%p A001045:= n -> (2^n-(-1)^n)/3;
%p A140944:= proc(n,k) if n = 0 then A001045(k); else procname(n-1,k+1)-procname(n-1,k) ; fi; end:
%p seq(seq(A140944(n,k),k=0..n),n=0..10); # _R. J. Mathar_, Sep 07 2009
%t T[0, 0]=0; T[1, 0]= T[0, 1]= 1; T[0, k_]:= T[0, k]= T[0, k-1] + 2*T[0, k-2]; T[n_, n_]=0; T[n_, k_]:= T[n, k] = T[n-1, k+1] - T[n-1, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* _Jean-François Alcover_, Dec 17 2014 *)
%t Table[2^k*(1-(-2)^(n-k))/3, {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 18 2023 *)
%o (PARI) T(n, k) = (2^k - 2^n*(-1)^(n+k))/3 \\ _Jianing Song_, Aug 11 2022
%o (Magma) [2^k*(1-(-2)^(n-k))/3: k in [0..n], n in [0..15]]; // _G. C. Greubel_, Feb 18 2023
%o (SageMath)
%o def A140944(n,k): return 2^k*(1 - (-2)^(n-k))/3
%o flatten([[A140944(n,k) for k in range(n+1)] for n in range(16)]) # _G. C. Greubel_, Feb 18 2023
%Y Cf. A000079, A001045, A016131, A045883, A052992, A083086, A084175.
%Y Cf. A091056, A097038, A109765, A115489, A140503, A192382, A246036.
%K sign,tabl,easy
%O 0,5
%A _Paul Curtz_, Jul 24 2008
%E Edited and extended by _R. J. Mathar_, Sep 07 2009
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