A301389 Decimal expansion of a constant derived from the differential equation L(x,y) = x * (1 + y*x*L'(x,y)) / (1 + x*L'(x,y)).
2, 8, 4, 5, 3, 4, 4, 9, 0, 3, 2, 0, 2, 5, 4, 7, 2, 1, 7, 2, 7, 7, 8, 4, 3, 3, 6, 2, 0, 9, 0, 5, 5, 7, 0, 9, 7, 6, 6, 1, 0, 3, 6, 1, 1, 4, 9, 4, 4, 1, 4, 3, 5, 1, 4, 5, 5, 9, 1, 7, 9, 6, 0, 6, 8, 5, 3, 4, 1, 9, 6, 7, 2, 7, 6, 4, 4, 2, 7, 0, 6, 2, 3, 7, 9, 8, 4, 8, 0, 2, 9, 5, 8, 9, 1, 7, 4, 0, 6, 5, 0, 8, 2, 1, 9
Offset: 1
Examples
Constant: t = 2.84534490320254721727784336209055709766103611494414\
35145591796068534196727644270623798480295891740650\
82199145990830799437319028986240326566511171283481\
64124258557293481255455658423617973823914494928144\
02390549176545225010379832242080737113084391329735\
50415116871947960699750963787855045833919347513162\
66142576403453447018744604133043007656376033045112\
82694417889680876297725614997044109272047404555831\
03178072390378468305666713745715501159999757302514\
68035430988987059516297431687872814834937381777727884...
DERIVATION.
Define L(x,y) by
L(x,y) = x * (1 + y*x*L'(x,y)) / (1 + x*L'(x,y))
where L'(x,y) = d/dx L(x,y).
Also let P(n,y) denote polynonmials in y of degree n-1 such that
L(x,y) = Sum_{n>=1} P(n,y) * x^n.
Then this constant t equals the limit of the largest real root of P(n,y) = 0 as n approaches infinity.
Thus, L(x, y >= t) is a power series in x that consists entirely of nonnegative coefficients of x^n for n>=1, while L(x, y < t) will have negative coefficients somewhere in the series.
See A301305 for a list of coefficients in L(x,y).
EXAMPLES.
At y = 3: if F(x) = x * (1 + 3*x*F'(x)) / (1 + x*F'(x)), then F(x) consists entirely of positive coefficients: F(x) = x + 2*x^2 + 6*x^3 + 22*x^4 + 94*x^5 + 474*x^6 + ...
At y = 2: if G(x) = x * (1 + 2*x*G'(x)) / (1 + x*G'(x)), then G(x) has negative coefficients: G(x) = x + x^2 + x^3 - 5*x^5 - 23*x^6 - 80*x^7 - 256*x^8 + ...
SPECIAL CASE.
At y = t, the coefficients of x^n in L(x, y = t) begin:
n=1: 1
n=2: 1.845344903202547217277843362090...
n=3: 4.965250720348689500886214520628...
n=4: 15.71195398585476459000000028776...
n=5: 54.45094229315423003059802798184...
n=6: 200.0279124924168961409663175538...
n=7: 765.2331026331498880864799981776...
n=8: 3016.273391007059589064353577450...
n=9: 12164.69846021846651010262473346...
n=10: 49958.6716816427739818467452330...
n=11: 208215.160884090085951262358449...
n=12: 878422.847138400462636146391350...
n=13: 3744079.94300463735553829872990...
n=14: 16098391.4459329863609360479668...
n=15: 69742236.3957366708693190076620...
n=16: 304134288.424775677905888472068...
n=17: 1333972959.55625603969319662400...
n=18: 5881057391.40269421518919556298...
n=19: 26046610024.3815379282982965072...
n=20: 115832831858.530165463839650824...
...
Series L(x,y) possesses the minimal nonnegative set of coefficients at y = t.
Incidentally, function L(x,y) satisfies
[x^n] exp(-n*L(x,y)) = ((y-1)*(n-1) - 1) [x^(n-1)] exp(-n*L(x,y)) for n>=1.
From _Vaclav Kotesovec_, Mar 22 2018: (Start)
Coefficient of [x^n] in L(x, y = t) is asymptotic to c * d^n / n^(3/2), where
d = 4.800584821563937430105758334563754815745511567342145151930777466927565...
c = 0.2473442204028460217878828954759832163023065757240699086838139... (End)
Crossrefs
Cf. A301305 (L(x,y)).