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A332776
a(n) = 1 + Sum_{k=1..n-1} binomial(n-1,k) * a(k) * a(n-k-1).
3
1, 1, 2, 5, 18, 83, 464, 3041, 22810, 192595, 1807328, 18658097, 210138882, 2563990859, 33691089824, 474327797585, 7123141539610, 113656386574099, 1920170741071280, 34242622099969217, 642792206343361602, 12669617513914228907, 261613287097165614224, 5647565141926833774977
OFFSET
0,3
LINKS
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