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

%I #8 Jun 28 2019 21:17:56

%S 1,-1,0,3,-6,-10,100,-175,-1470,11214,-4032,-447678,2813580,8767044,

%T -254393568,1156311585,14213048850,-237139066450,423094740640,

%U 26925567437054,-323136231452892,-998293111680228,67449022208054760,-562713810943757746,-7585754355598687268,220643947556639812100

%N Expansion of e.g.f. 1 / (BesselI(0,2*x) + BesselI(1,2*x)).

%C E.g.f. is inverse of e.g.f. for A001405.

%F E.g.f.: 1 / Sum_{k>=0} binomial(k,floor(k/2))*x^k/k!.

%t nmax = 25; CoefficientList[Series[1/(BesselI[0, 2 x] + BesselI[1, 2 x]), {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n, k] Binomial[k, Floor[k/2]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 25}]

%Y Cf. A001405, A308850.

%K sign

%O 0,4

%A _Ilya Gutkovskiy_, Jun 28 2019