A326177 Denominators of a recurrence relation arising in impact dynamics.

1, 2, 2, 20, 5, 700, 350, 7000, 1750, 215600, 215600, 12512500, 350350000, 7007000000, 1001000000, 45815000000, 148898750000, 121989767900000000, 121989767900000000, 30497441975000000, 4879590716000000, 5106491684294000000000, 464226516754000000000

Offset

0,2

Links

Table of n, a(n) for n=0..22.

André S. Carvalho, Jorge M. Martins, Exact restitution and generalizations for the Hunt-Crossley contact model, Mechanism and Machine Theory, (2019) 139, 174-194.

Formula

Initial terms are

B(0) = 0,

B(1) = -3/2.

Subsequent terms are computed from

B(n) = 1/(2*(n+2)) * ( (6-7*n)*B(n-1) - 3*(n-1)*B(n-2) + 2*r(n) ),

where r(n) denotes a finite sum given by

r(n) = Sum_{j=2..n-1} B(j)*( (n-j)*B(n-j-1) + (3*n-3*j+1)*B(n-j) + 2*(n-j+1)*B(n-j+1) ).

Finally, the present sequence is given by the denominators of B(n), which is employed to compute the inverse restitution, through an infinite sum, given by

y(x) = Sum_{n=1..oo} B(n)*(x-1)^n.

Mathematica

Denominator@With[{m = 22}, Module[{B}, Join[{B[0] = 0}, {B[1] = -3/2}, Table[B[n] = 1/(2 (n + 2)) ((6 - 7 n) B[n - 1] - 3 (n - 1) B[n - 2] + 2 Sum[B[j] ((n - j) B[n - j - 1] + (3 n - 3 j + 1) B[n - j] + 2 (n - j + 1) B[n - j + 1]), {j, 2, n - 1}]), {n, 2, m}]]]]

Crossrefs

Cf. A326176 (numerators).

Sequence in context: A172194 A338400 A093777 * A370709 A103129 A322898

Adjacent sequences: A326174 A326175 A326176 * A326178 A326179 A326180

Keyword

nonn,frac

Author

André S. Carvalho, Jun 11 2019

Status

approved