A202668 G.f. satisfies: A(x) = exp( Sum_{n>=1} (A(x) - (-1)^n)^n * x^n/n ).
1, 2, 4, 12, 42, 158, 618, 2498, 10360, 43832, 188420, 820608, 3613212, 16057640, 71933768, 324482500, 1472604586, 6719100254, 30804229858, 141829955338, 655541387406, 3040527731790, 14147444737654, 66018910398574, 308898542610666, 1448867831911170
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 42*x^4 + 158*x^5 + 618*x^6 + ... where log(A(x)) = (A(x) + 1)*x + (A(x) - 1)^2*x^2/2 + (A(x) + 1)^3*x^3/3 + (A(x) - 1)^4*x^4/4 + ... log( A(x)*(1-x*A(x)) ) = 1/(1 + x*A(x))*x + 1/(1 - x*A(x))^2*x^2/2 + 1/(1 + x*A(x))^3*x^3/3 + 1/(1 - x*A(x))^4*x^4/4 + ... From _Paul D. Hanna_, Oct 11 2024: (Start) SPECIFIC VALUES. A(t) = 2 at t = 0.195782060076367892865630673522992184838101... where 12*t^3 - 4*t^2 - 15*t + 3 = 0. A(t) = 3/2 at t = 0.1528468026979892250300352740045422934687... where 45*t^3 - 18*t^2 - 260*t + 40 = 0. A(1/6) = 1.5975588141693553913621853542774164447766461118908... A(1/7) = 1.4422077780342017637064340698606478883307441400444... A(1/8) = 1.3558965312086216338851741626422486193364696459775... A(1/9) = 1.2992876417963412242026519185070094965390617289384... A(1/10) = 1.258828814568496961617240364573696812116531654741... (End)
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..800
- Vaclav Kotesovec, Recurrence (of order 12)
Programs
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PARI
{a(n) = my(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (A - (-1)^m +x*O(x^n))^m * x^m/m))); polcoeff(A, n)} for(n=0,30,print1(a(n),", "))
Formula
G.f. satisfies: A(x) = 1/(1-x*A(x)) * exp( Sum_{n>=1} 1/(1 - (-1)^n*x*A(x))^n * x^n/n ).
G.f. satisfies: A(x) = sqrt( (1 - (A(x)+1)^2*x^2)/(1 - (A(x)-1)^2*x^2) ) / (1 - (A(x)+1)*x).
G.f. satisfies: 0 = -(1+x) - x*A(x) + (1+x)*(1-x)^2*A(x)^2 - x*(1-x)^2*A(x)^3 - x^2*(1+x)*A(x)^4 + x^3*A(x)^5.
From Vaclav Kotesovec, Oct 11 2024: (Start)
a(n) ~ sqrt((-1 - s + (-1 - 2*r + 3*r^2)*s^2 + (-1 + 4*r - 3*r^2)*s^3 - r*(2 + 3*r)*s^4 + 3*r^2*s^5)/(1 - r*(1 + 3*s) + r^2*(-1 + 6*s - 6*s^2) + r^3*(1 - 3*s - 6*s^2 + 10*s^3))) / (2*sqrt(Pi) * n^(3/2) * r^(n - 1/2)), where r = 0.20089689587759865228481815120918189691453519374477284069915... and s = 2.3487742728380350386577466365052703249852809669846393564277... are positive real roots of the system of equations s^2*(1 + r^3*(-1 + s)^2*(1 + s)) = 1 + r^2*(-1 + s)^2*s^2 + r*(1 + s + s^2 + s^3) and 2*(-1 + r)^2*(1 + r)*s + 5*r^3*s^4 = r*(1 + 3*(-1 + r)^2*s^2 + 4*r*(1 + r)*s^3).
Numerically, a(n) ~ c * d^n / n^(3/2), where d = 1/r = 4.977677706923229216140896605827075562322447814212438341196056039... and c = 0.7100736662419384614471705442776864037581200760804364785319... (End)