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A123956 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.

%I A123956 #38 Apr 25 2025 21:11:31
%S A123956 -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 A123956 48,-32,-32,1,-7,24,56,-80,-112,64,64,-1,8,32,-80,-160,192,256,-128,
%U A123956 -128,1,9,-40,-120,240,432,-448,-576,256,256,-1,-10,-50,160,400,-672,-1120,1024,1280,-512,-512
%N A123956 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 A123956 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 A123956 CRC Standard Mathematical Tables and Formulae, 16th ed. 1996, p. 484.
%D A123956 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 A123956 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 A123956 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
%H A123956 Alastair MacDougall, <a href="https://doi.org/10.2307/3619060">83.31 A Pascal-like triangle for coefficients of Chebyshev polynomials">, The Mathematical Gazette, Vol. 83, Issue 497 (Jul 1999), pp. 276-280.
%F A123956 From _M. F. Hasler_, Nov 30 2022: (Start)
%F A123956 a(n,0) = (-1)^(n+1), a(n,1) = (-1)^floor(n/2)*n,
%F A123956 a(n,2) = (-1)^floor((n+1)/2)*A007590(n) = (-1)^floor((n+1)/2)*floor(n^2 / 2),
%F A123956 a(n,n) = a(n,n-1) = (-2)^(n-1) (n > 0),
%F A123956 a(n,3) / a(n,2) = { n/3 if n odd, -4*(n+2)/n if n even },
%F A123956 a(n,4) / a(n,3) = n/4 if n is even. (End)
%F A123956 From _Peter Bala_, Feb 06 2025: (Start)
%F A123956 Let T(n, x) and U(n, x) denote the n-th Chebyshev polynomial of the first and second kind. It appears that the row g.f.'s are as follows: for n >= 0,
%F A123956 row 4*n+1: T(4*n+1, x) + U(4*n, x); row 4*n+2: - 2 - T(4*n+2, x) - U(4*n+1, x);
%F A123956 row 4*n+3: 2 + T(4*n+3, x) + U(4*n+2, x); row 4*n+4: - T(4*n+4, x) - U(4*n+3, x). (End)
%e A123956 Triangle begins:
%e A123956   {-1},
%e A123956   { 1,   1},
%e A123956   {-1,  -2,  -2},
%e A123956   { 1,  -3,   4,    4},
%e A123956   {-1,   4,   8,   -8,   -8},
%e A123956   { 1,   5, -12,  -20,   16,   16},
%e A123956   {-1,  -6, -18,   32,   48,  -32,   -32},
%e A123956   { 1,  -7,  24,   56,  -80, -112,    64,   64},
%e A123956   {-1,   8,  32,  -80, -160,  192,   256, -128, -128},
%e A123956   { 1,   9, -40, -120,  240,  432,  -448, -576,  256,  256},
%e A123956   {-1, -10, -50,  160,  400, -672, -1120, 1024, 1280, -512, -512},
%e A123956   ...
%t A123956 p[ -1, x] = 0; p[0, x] = 1; p[1, x] = x + 1;
%t A123956 p[k_, x_] := p[k, x] = 2*x*p[k - 1, x] - p[k - 2, x];
%t A123956 w = Table[CoefficientList[p[n, x], x], {n, 0, 10}];
%t A123956 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 A123956 b = Table[CoefficientList[ExpandAll[y^(d - 1)*(CharacteristicPolynomial[An[d], x] /. x -> 1/y)] /. 1/y -> 1, y], {d, 1, 11}] // Flatten
%o A123956 (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 A123956 Cf. A008312, A028297, A053117, A123235.
%K A123956 sign,uned,tabl
%O A123956 0,5
%A A123956 _Roger L. Bagula_ and _Gary W. Adamson_, Oct 27 2006
%E A123956 Offset changed to 0 by _M. F. Hasler_, Nov 30 2022