login
Expansion of e.g.f. exp(-x) / BesselI(0,2*x).
1

%I #11 Jul 03 2019 04:08:10

%S 1,-1,-1,5,7,-71,-139,2071,5335,-103207,-331511,7853251,30256381,

%T -847377805,-3808492297,123081031165,632196102455,-23155450005175,

%U -133802756269735,5477371955388355,35167483918412257,-1591161899246627297,-11237664710770159597,556875003328690925825,4290500676272573740429

%N Expansion of e.g.f. exp(-x) / BesselI(0,2*x).

%C E.g.f. is inverse of e.g.f. for A002426 (central trinomial coefficients).

%F E.g.f.: 1 / Sum_{k>=0} A002426(k)*x^k/k!.

%t nmax = 24; CoefficientList[Series[Exp[-x]/BesselI[0, 2 x], {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n, k] 3^k Hypergeometric2F1[1/2, -k, 1, 4/3] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]

%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(-x) / besseli(0,2*x))) \\ _Michel Marcus_, Jul 02 2019

%Y Cf. A002426, A167022, A308847.

%K sign

%O 0,4

%A _Ilya Gutkovskiy_, Jun 28 2019