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A326176
Numerators of a recurrence relation arising in impact dynamics.
1
0, -3, 3, -27, 6, -759, 351, -6639, 1599, -192537, 190221, -10993971, 308346453, -6203405379, 894162999, -41382469053, 136200425703, -113114950247136633, 114732379536286977, -29101190167836903, 4724140398296547, -5014991521798555929939, 462302451532498523799
OFFSET
0,2
LINKS
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 numerators 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
Numerator@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. A326177 (denominators).
Sequence in context: A355558 A363471 A303822 * A157036 A080302 A080272
KEYWORD
sign,frac
AUTHOR
André S. Carvalho, Jun 11 2019
STATUS
approved