A173435
Inverse binomial transform of A143025, assuming offset zero there.
Original entry on oeis.org
8, -6, 12, -25, 52, -106, 212, -420, 832, -1656, 3312, -6640, 13312, -26656, 53312, -106560, 212992, -425856, 851712, -1703680, 3407872, -6816256, 13632512, -27264000, 54525952, -109049856, 218099712, -436203520, 872415232, -1744838656, 3489677312, -6979338240
Offset: 0
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Join[{8},LinearRecurrence[{-4,-6,-4},{-6,12,-25},40]] (* Harvey P. Dale, Sep 25 2013 *)
A191897
Coefficients of the Z(n,x) polynomials; Z(0,x) = 1, Z(1,x) = x and Z(n,x) = x*Z(n-1,x) - 2*Z(n-2,x), n >= 2.
Original entry on oeis.org
1, 1, 0, 1, 0, -2, 1, 0, -4, 0, 1, 0, -6, 0, 4, 1, 0, -8, 0, 12, 0, 1, 0, -10, 0, 24, 0, -8, 1, 0, -12, 0, 40, 0, -32, 0, 1, 0, -14, 0, 60, 0, -80, 0, 16, 1, 0, -16, 0, 84, 0, -160, 0, 80, 0, 1, 0, -18, 0, 112, 0, -280, 0, 240, 0, -32
Offset: 0
The first few rows of the coefficients of the Z(n,x) are
1;
1, 0;
1, 0, -2;
1, 0, -4, 0;
1, 0, -6, 0, 4;
1, 0, -8, 0, 12, 0;
1, 0, -10, 0, 24, 0, -8;
1, 0, -12, 0, 40, 0, -32, 0;
1, 0, -14, 0, 60, 0, -80, 0, 16;
1, 0, -16, 0, 84, 0, -160, 0, 80, 0;
Row sum without sign:
A113405(n+1).
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nmax:=10: Z(0, x):=1 : Z(1, x):=x: for n from 2 to nmax do Z(n, x) := x*Z(n-1, x) - 2*Z(n-2, x) od: for n from 0 to nmax do for k from 0 to n do T(n, k) := coeff(Z(n, x), x, n-k) od: od: seq(seq(T(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Jun 27 2011, revised Nov 29 2012
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a[n_, k_] := If[OddQ[k], 0, 2^(k/2)*Coefficient[ ChebyshevU[n, x/2], x, n-k]]; Flatten[ Table[ a[n, k], {n, 0, 10}, {k, 0, n}]] (* Jean-François Alcover, Aug 02 2012, from 2nd formula *)
A373392
Inverse binomial transform of A135318.
Original entry on oeis.org
1, 0, 0, 1, -2, 3, -4, 7, -18, 51, -136, 339, -814, 1935, -4620, 11111, -26842, 64923, -156944, 379067, -915078, 2208711, -5331476, 12870639, -31072754, 75018195, -181113240, 437248771, -1055610782, 2548462143, -6152518684, 14853483127, -35859484938, 86572485771
Offset: 0
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LinearRecurrence[{-4, -5, -2, 2}, {1, 0, 0, 1}, 35] (* Amiram Eldar, Jun 09 2024 *)
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a(n) = ((-([-2,-1;-1, 0]^(n-2))[2, 1]) - 2*((I-1)^(n-4) + (-I-1)^(n-4)))/3; \\ Thomas Scheuerle, Jun 04 2024
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