A292927 - OEIS

%I #8 Oct 10 2017 07:24:52

%S 1,1,3,11,52,258,1343,7257,40275,228278,1315922,7691196,45473095,

%T 271482064,1634359974,9910367591,60474714189,371087272878,

%U 2288372703482,14174212020218,88145412404781,550128210470715,3444680265887877,21633697884627237,136238869051956545,860130515526195618,5442975808350831237,34517730741744461395,219338548934112758627

%N G.f. A(x) satisfies: A( x^2*A(x) - x*A(x)^3 ) = x^3.

%H Paul D. Hanna, <a href="/A292927/b292927.txt">Table of n, a(n) for n = 1..520</a>

%F a(n) ~ c * d^n / n^(3/2), where d = 6.705143079646414499260567437823218217... and c = 0.03594678018676382296451433... - _Vaclav Kotesovec_, Oct 10 2017

%e G.f.: A(x) = x + x^2 + 3*x^3 + 11*x^4 + 52*x^5 + 258*x^6 + 1343*x^7 + 7257*x^8 + 40275*x^9 + 228278*x^10 + 1315922*x^11 + 7691196*x^12 + 45473095*x^13 + 271482064*x^14 + 1634359974*x^15 + 9910367591*x^16 +...

%e such that A( x^2*A(x) - x*A(x)^3 ) = x^3.

%e RELATED SERIES.

%e x^2*A(x) - x*A(x)^3 = x^3 - x^6 - x^9 - x^12 - 8*x^15 - 13*x^18 - 37*x^21 - 159*x^24 - 388*x^27 - 1403*x^30 - 5090*x^33 - 15931*x^36 - 58532*x^39 +...

%e Let B(x) be the series reversion of A(x), so that B(A(x)) = x, then

%e B(x) = x - x^2 - x^3 - x^4 - 8*x^5 - 13*x^6 - 37*x^7 - 159*x^8 - 388*x^9 - 1403*x^10 - 5090*x^11 - 15931*x^12 - 58532*x^13 - 207536*x^14 - 719812*x^15 - 2641077*x^16 - 9504900*x^17 - 34393816*x^18 - 126750932*x^19 - 464389638*x^20 +...

%e then x^2*A(x) - x*A(x)^3 = B(x^3).

%o (PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -Vec(subst(F, x, x^2*F - x*F^3))[#A] ); polcoeff(A, n)}

%o for(n=1, 40, print1(a(n), ", "))

%Y Cf. A265940, A268039, A272463, A292928.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Sep 26 2017