OFFSET
0,5
COMMENTS
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
REFERENCES
CRC Standard Mathematical Tables and Formulae, 16th ed. 1996, p. 484.
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.
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Lutterbach, Approximating ODE y′=f(t,y) by using leapfrog method, Mathematics Stack Exchange, Nov 21 2019
FORMULA
From M. F. Hasler, Nov 30 2022: (Start)
a(n,0) = (-1)^(n+1), a(n,1) = (-1)^floor(n/2)*n,
a(n,2) = (-1)^floor((n+1)/2)*A007590(n) = (-1)^floor((n+1)/2)*floor(n^2 / 2),
a(n,n) = a(n,n-1) = (-2)^(n-1) (n > 0),
a(n,3) / a(n,2) = { n/3 if n odd, -4*(n+2)/n if n even },
a(n,4) / a(n,3) = n/4 if n is even. (End)
EXAMPLE
Triangle begins:
{-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, 48, -32, -32},
{ 1, -7, 24, 56, -80, -112, 64, 64},
{-1, 8, 32, -80, -160, 192, 256, -128, -128},
{ 1, 9, -40, -120, 240, 432, -448, -576, 256, 256},
{-1, -10, -50, 160, 400, -672, -1120, 1024, 1280, -512, -512},
...
MATHEMATICA
p[ -1, x] = 0; p[0, x] = 1; p[1, x] = x + 1;
p[k_, x_] := p[k, x] = 2*x*p[k - 1, x] - p[k - 2, x];
w = Table[CoefficientList[p[n, x], x], {n, 0, 10}];
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}];
b = Table[CoefficientList[ExpandAll[y^(d - 1)*(CharacteristicPolynomial[An[d], x] /. x -> 1/y)] /. 1/y -> 1, y], {d, 1, 11}] // Flatten
PROG
(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
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula and Gary W. Adamson, Oct 27 2006
EXTENSIONS
Offset changed to 0 by M. F. Hasler, Nov 30 2022
STATUS
approved