%I #12 Jul 03 2019 04:08:03
%S 1,-2,2,4,-14,-52,284,1496,-10958,-74372,681652,5656616,-62226116,
%T -610306712,7832965352,88645228304,-1300254163918,-16676932459172,
%U 275196007522436,3944890321174664,-72330003541955564,-1145979961718846152,23112838345877865752,401070175407076000624,-8824400094691629670724
%N Expansion of e.g.f. exp(-2*x) / BesselI(0,2*x).
%C E.g.f. is inverse of e.g.f. for A000984 (central binomial coefficients).
%F E.g.f.: 1 / Sum_{k>=0} binomial(2*k,k)*x^k/k!.
%t nmax = 24; CoefficientList[Series[Exp[-2 x]/BesselI[0, 2 x], {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n, k] Binomial[2 k, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(-2*x) / besseli(0,2*x))) \\ _Michel Marcus_, Jul 02 2019
%Y Cf. A000984, A002420, A178955, A308848.
%K sign
%O 0,2
%A _Ilya Gutkovskiy_, Jun 28 2019