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High-energy theory

Checking a collaborator's series

About 14 min if you have seen this material before, 35 min or so if it is new to you.

Background
Sometimes a tedious part of the job in theoretical physics is finding mistakes in a paper, or from a collaborator, or your own. This problem is an example of that kind of work, in a case where a pattern in the numbers lets you check them without recomputing each one. Patterns like these show up with some frequency in the equations physics deals with. The series below is the same one as in the previous problem: order nn is Dn(−λ)n/n!D_n(-\lambda)^n/n!, summed over the DnD_n ways of pairing the 4n4n line-ends of the nn four-ended vertices of a A picture standing for one contribution to a calculation, with vertices where particles meet and lines joining them..
A collaborator has computed the series ∑n≥0Dn (−λ)nn!,Dn=(4n−1)!!,\sum_{n \ge 0} \frac{D_n\,(-\lambda)^n}{n!}, \qquad D_n = (4n-1)!!, at λ=0.01\lambda = 0.01, and reports these magnitudes Tn=Dnλn/n!T_n = D_n\lambda^n/n!. Exactly one entry is wrong. 
    n    T_n
   ---   -----------
    0    1.00000
    1    0.0300000
    2    0.00480000
    3    0.00173250
    4    0.000844594
    5    0.000545608
    6    0.000439214
    7    0.000423528
    8    0.000475939
    9    0.000610789
   10    0.000881368
The idea is to find the wrong entry without computing every term, and then say what the collaborator did instead of the calculation they meant to do.
(a)
Consecutive terms of this series have a ratio you can write down in closed form from DnD_n alone. Work it out, then test the table against it. Give every nn for which the table's Tn+1/TnT_{n+1}/T_n disagrees with the rule, as whole numbers separated by spaces.
(b)
The bad entry was computed with the wrong value of DnD_n. What number did the collaborator use in its place?

Past 1 hr on this? That usually means a missing prerequisite rather than a lack of effort — worth reading up before pushing further, starting with Asymptotic expansion (Wikipedia).

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