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A330794 Inverse of the Jacobsthal triangle (A322942). Triangle read by rows, T(n, k) for 0 <= k <= n.

%I A330794 #18 Sep 28 2023 04:08:16
%S A330794 1,-1,1,1,-2,1,-1,1,-3,1,1,4,2,-4,1,-1,-7,10,4,-5,1,1,-14,-25,16,7,-6,
%T A330794 1,-1,65,-21,-55,21,11,-7,1,1,-24,196,-8,-98,24,16,-8,1,-1,-367,-204,
%U A330794 400,42,-154,24,22,-9,1,1,774,-963,-688,666,148,-222,20,29,-10,1
%N A330794 Inverse of the Jacobsthal triangle (A322942). Triangle read by rows, T(n, k) for 0 <= k <= n.
%C A330794 The inverse matrix of the Riordan square (cf. A321620) generated by (1 - 2*x^2)/((1 + x)*(1 - 2*x)).
%H A330794 G. C. Greubel, <a href="/A330794/b330794.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A330794 G. C. Greubel, <a href="/A330794/a330794.txt">SageMath code</a>
%F A330794 From _G. C. Greubel_, Sep 15 2023: (Start)
%F A330794 T(n, 0) = (-1)^n.
%F A330794 T(n, n) = 1.
%F A330794 T(n, n-1) = -n.
%F A330794 T(n, n-2) = A152947(n-1). (End)
%e A330794 Triangle starts:
%e A330794 [0]   1;
%e A330794 [1]  -1,    1;
%e A330794 [2]   1,   -2,    1;
%e A330794 [3]  -1,    1,   -3,    1;
%e A330794 [4]   1,    4,    2,   -4,    1;
%e A330794 [5]  -1,   -7,   10,    4,   -5,    1;
%e A330794 [6]   1,  -14,  -25,   16,    7,   -6,    1;
%e A330794 [7]  -1,   65,  -21,  -55,   21,   11,   -7,    1;
%e A330794 [8]   1,  -24,  196,   -8,  -98,   24,   16,   -8,    1;
%e A330794 [9]  -1, -367, -204,  400,   42, -154,   24,   22,   -9,    1;
%t A330794 m=30;
%t A330794 A322942:= CoefficientList[CoefficientList[Series[(1-2*t^2)/(1-(x+1)*t-2*t^2), {x,0,m}, {t,0,m}], t], x];
%t A330794 M:= M= Table[If[k<=n, A322942[[n+1,k+1]], 0], {n,0,m}, {k,0,m}];
%t A330794 g:= g= Inverse[M];
%t A330794 A330794[n_, k_]:= g[[n+1,k+1]];
%t A330794 Table[A330794[n,k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Sep 20 2023 *)
%o A330794 (Sage) # uses[riordan_array from A256893]
%o A330794 Jacobsthal = (2*x^2 - 1)/((x + 1)*(2*x - 1))
%o A330794 riordan_array(Jacobsthal, Jacobsthal, 10).inverse()
%Y A330794 Cf. A152947, A256893, A321620, A322942.
%K A330794 sign,tabl
%O A330794 0,5
%A A330794 _Peter Luschny_, Jan 03 2020