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A330794
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Inverse of the Jacobsthal triangle (A322942). Triangle read by rows, T(n, k) for 0 <= k <= n.
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1
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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
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OFFSET
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0,5
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COMMENTS
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The inverse matrix of the Riordan square (cf. A321620) generated by (1 - 2*x^2)/((1 + x)*(1 - 2*x)).
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LINKS
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FORMULA
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T(n, 0) = (-1)^n.
T(n, n) = 1.
T(n, n-1) = -n.
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EXAMPLE
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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;
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MATHEMATICA
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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];
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PROG
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(Sage) # uses[riordan_array from A256893]
Jacobsthal = (2*x^2 - 1)/((x + 1)*(2*x - 1))
riordan_array(Jacobsthal, Jacobsthal, 10).inverse()
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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