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%I #30 Dec 19 2022 04:33:13
%S -1,1,1,-1,-2,-2,1,-3,4,4,-1,4,8,-8,-8,1,5,-12,-20,16,16,-1,-6,-18,32,
%T 48,-32,-32,1,-7,24,56,-80,-112,64,64,-1,8,32,-80,-160,192,256,-128,
%U -128,1,9,-40,-120,240,432,-448,-576,256,256,-1,-10,-50,160,400,-672,-1120,1024,1280,-512,-512
%N Triangle of coefficients of polynomials P(k) = 2*X*P(k - 1) - P(k - 2), P(0) = -1, P(1) = 1 + X, with twisted signs.
%C Up to signs also the coefficients of polynomials y(n+1) = y(n-1) - 2*h*y(n), arising when the ODE y' = -y is numerically solved with the leapfrog (a.k.a. two-step Nyström) method, with y(0) = 1, y(1) = 1 - h. In this case, the coefficients are negative exactly for the odd powers of h. - _M. F. Hasler_, Nov 30 2022
%D CRC Standard Mathematical Tables and Formulae, 16th ed. 1996, p. 484.
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Lutterbach, <a href="https://math.stackexchange.com/questions/3444485/approximating-ode-y-ft-y-by-using-leapfrog-method">Approximating ODE y′=f(t,y) by using leapfrog method</a>, Mathematics Stack Exchange, Nov 21 2019
%F From _M. F. Hasler_, Nov 30 2022: (Start)
%F a(n,0) = (-1)^(n+1), a(n,1) = (-1)^floor(n/2)*n,
%F a(n,2) = (-1)^floor((n+1)/2)*A007590(n) = (-1)^floor((n+1)/2)*floor(n^2 / 2),
%F a(n,n) = a(n,n-1) = (-2)^(n-1) (n > 0),
%F a(n,3) / a(n,2) = { n/3 if n odd, -4*(n+2)/n if n even },
%F a(n,4) / a(n,3) = n/4 if n is even. (End)
%e Triangle begins:
%e {-1},
%e { 1, 1},
%e {-1, -2, -2},
%e { 1, -3, 4, 4},
%e {-1, 4, 8, -8, -8},
%e { 1, 5, -12, -20, 16, 16},
%e {-1, -6, -18, 32, 48, -32, -32},
%e { 1, -7, 24, 56, -80, -112, 64, 64},
%e {-1, 8, 32, -80, -160, 192, 256, -128, -128},
%e { 1, 9, -40, -120, 240, 432, -448, -576, 256, 256},
%e {-1, -10, -50, 160, 400, -672, -1120, 1024, 1280, -512, -512},
%e ...
%t p[ -1, x] = 0; p[0, x] = 1; p[1, x] = x + 1;
%t p[k_, x_] := p[k, x] = 2*x*p[k - 1, x] - p[k - 2, x];
%t w = Table[CoefficientList[p[n, x], x], {n, 0, 10}];
%t An[d_] := Table[If[n == d && m <n, -w[[n]][[d - m + 1]], If[m == n + 1, 1, 0]], {n, 1, d}, {m, 1, d}];
%t b = Table[CoefficientList[ExpandAll[y^(d - 1)*(CharacteristicPolynomial[An[d], x] /. x -> 1/y)] /. 1/y -> 1, y], {d, 1, 11}] // Flatten
%o (PARI) P=List([-1,1-'x]); {A123956(n,k)=for(i=#P, n+1, listput(P, P[i-1]-2*'x*P[i])); polcoef(P[n+1],k)*(-1)^((n-k-1)\2+!k*n\2)} \\ _M. F. Hasler_, Nov 30 2022
%Y Cf. A123235.
%K uned,tabl,sign
%O 0,5
%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 27 2006
%E Offset changed to 0 by _M. F. Hasler_, Nov 30 2022
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