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Vaclav Kotesovec, <a href="/A335441/b335441.txt">Table of n, a(n) for n = 0..470</a>

From Vaclav Kotesovec, Jun 11 2020: (Start)

E.g.f.: (BesselY(0, sqrt(2))*(BesselJ(1, sqrt(2)*exp(x/2)) - sqrt(2)*exp(x/2)*BesselJ(0, sqrt(2)*exp(x/2))) + BesselJ(0, sqrt(2))*(sqrt(2)*exp(x/2)*BesselY(0, sqrt(2)*exp(x/2)) - BesselY(1, sqrt(2)*exp(x/2)))) / (BesselJ(1, sqrt(2)*exp(x/2))*BesselY(0, sqrt(2)) - BesselJ(0, sqrt(2))*BesselY(1, sqrt(2)*exp(x/2))).

a(n) ~ 2 * n! / r^(n+1), where r = 1.35169030867903432729790416904526340210784862703704233748118252928787... is the smallest real root of the equation BesselY(0, sqrt(2))*BesselJ(1, sqrt(2)*exp(r/2)) = BesselJ(0, sqrt(2))*BesselY(1, sqrt(2)*exp(r/2)). (End)

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a(n) = 1 + Sum_{k=1..n-1} binomial(n-2,k-1) * a(k) * a(n-k-1).

1, 1, 2, 4, 11, 40, 176, 907, 5360, 35668, 263789, 2146390, 19054040, 183248581, 1897952690, 21061861828, 249309196559, 3135518918800, 41754612283244, 586922460056851, 8684272948653068, 134919751191875572, 2195942678525060093, 37365571515146318650

0,3

E.g.f. A(x) satisfies: A''(x) = exp(x) + A(x) * A'(x).

a[n_] := a[n] = 1 + Sum[Binomial[n - 2, k - 1] 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[Integrate[Exp[x] + A[x] D[A[x], x], x], x] + O[x]^(terms + 1)], terms]; CoefficientList[A[x], x] Range[0, terms]!

Cf. A000111, A006677, A052886, A054687, A080635, A332776.

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Ilya Gutkovskiy, Jun 10 2020

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