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

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, 1, -1, 65, -21, -55, 21, 11, -7, 1, 1, -24, 196, -8, -98, 24, 16, -8, 1, -1, -367, -204, 400, 42, -154, 24, 22, -9, 1, 1, 774, -963, -688, 666, 148, -222, 20, 29, -10, 1

Offset: 0

Author

Peter Luschny, Jan 03 2020

Keywords

• sign
• tabl

Comments

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

Examples

			Triangle starts:
[0]   1;
[1]  -1,    1;
[2]   1,   -2,    1;
[3]  -1,    1,   -3,    1;
[4]   1,    4,    2,   -4,    1;
[5]  -1,   -7,   10,    4,   -5,    1;
[6]   1,  -14,  -25,   16,    7,   -6,    1;
[7]  -1,   65,  -21,  -55,   21,   11,   -7,    1;
[8]   1,  -24,  196,   -8,  -98,   24,   16,   -8,    1;
[9]  -1, -367, -204,  400,   42, -154,   24,   22,   -9,    1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• G. C. Greubel, SageMath code

Crossrefs

Cf. A152947, A256893, A321620, A322942.

Programs

Mathematica

m=30;
A322942:= CoefficientList[CoefficientList[Series[(1-2*t^2)/(1-(x+1)*t-2*t^2), {x,0,m}, {t,0,m}], t], x];
M:= M= Table[If[k<=n, A322942[[n+1,k+1]], 0], {n,0,m}, {k,0,m}];
g:= g= Inverse[M];
A330794[n_, k_]:= g[[n+1,k+1]];
Table[A330794[n,k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 20 2023 *)

Sage

# uses[riordan_array from A256893]
Jacobsthal = (2*x^2 - 1)/((x + 1)*(2*x - 1))
riordan_array(Jacobsthal, Jacobsthal, 10).inverse()

Formula

From G. C. Greubel, Sep 15 2023: (Start)

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

T(n, n) = 1.

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

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