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Inverse of the Jacobsthal triangle (A322942). Triangle read by rows, T(n, k) for 0 <= k <= n.
1

%I #18 Sep 28 2023 04:08:16

%S 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 1,-1,65,-21,-55,21,11,-7,1,1,-24,196,-8,-98,24,16,-8,1,-1,-367,-204,

%U 400,42,-154,24,22,-9,1,1,774,-963,-688,666,148,-222,20,29,-10,1

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

%C The inverse matrix of the Riordan square (cf. A321620) generated by (1 - 2*x^2)/((1 + x)*(1 - 2*x)).

%H G. C. Greubel, <a href="/A330794/b330794.txt">Rows n = 0..50 of the triangle, flattened</a>

%H G. C. Greubel, <a href="/A330794/a330794.txt">SageMath code</a>

%F From _G. C. Greubel_, Sep 15 2023: (Start)

%F T(n, 0) = (-1)^n.

%F T(n, n) = 1.

%F T(n, n-1) = -n.

%F T(n, n-2) = A152947(n-1). (End)

%e Triangle starts:

%e [0] 1;

%e [1] -1, 1;

%e [2] 1, -2, 1;

%e [3] -1, 1, -3, 1;

%e [4] 1, 4, 2, -4, 1;

%e [5] -1, -7, 10, 4, -5, 1;

%e [6] 1, -14, -25, 16, 7, -6, 1;

%e [7] -1, 65, -21, -55, 21, 11, -7, 1;

%e [8] 1, -24, 196, -8, -98, 24, 16, -8, 1;

%e [9] -1, -367, -204, 400, 42, -154, 24, 22, -9, 1;

%t m=30;

%t A322942:= CoefficientList[CoefficientList[Series[(1-2*t^2)/(1-(x+1)*t-2*t^2), {x,0,m}, {t,0,m}], t], x];

%t M:= M= Table[If[k<=n, A322942[[n+1,k+1]], 0], {n,0,m}, {k,0,m}];

%t g:= g= Inverse[M];

%t A330794[n_, k_]:= g[[n+1,k+1]];

%t Table[A330794[n,k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Sep 20 2023 *)

%o (Sage) # uses[riordan_array from A256893]

%o Jacobsthal = (2*x^2 - 1)/((x + 1)*(2*x - 1))

%o riordan_array(Jacobsthal, Jacobsthal, 10).inverse()

%Y Cf. A152947, A256893, A321620, A322942.

%K sign,tabl

%O 0,5

%A _Peter Luschny_, Jan 03 2020