OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..454
FORMULA
E.g.f. A(x) satisfies: d/dx A(x) = exp(x) + A(x) * (A(x) - 1).
From Vaclav Kotesovec, Jun 09 2020: (Start)
E.g.f.: exp(x/2) * (BesselJ(2, 2*exp(x/2)) * BesselY(0,2) - BesselJ(0,2) * BesselY(2, 2*exp(x/2))) / (BesselJ(1, 2*exp(x/2)) * BesselY(0,2) - BesselJ(0,2) * BesselY(1, 2*exp(x/2))).
a(n) ~ n! / r^(n+1), where r = 1.0654335847261788612657252860730850911833168584... is the smallest real root of the equation BesselJ(1, 2*exp(r/2)) * BesselY(0,2) = BesselJ(0,2) * BesselY(1, 2*exp(r/2)). (End)
MATHEMATICA
a[n_] := a[n] = 1 + Sum[Binomial[n - 1, k] a[k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]
terms = 23; A[_] = 0; Do[A[x_] = Normal[Integrate[Exp[x] + A[x] (A[x] - 1), x] + O[x]^(terms + 1)], terms]; CoefficientList[A[x], x] Range[0, terms]!
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jun 08 2020
STATUS
approved