A268039 - OEIS

%I #17 May 13 2024 18:22:39

%S 1,1,2,9,40,192,959,4988,26587,144672,800229,4486914,25444355,

%T 145681030,840988369,4889658927,28607653570,168299372745,994978254267,

%U 5908150711835,35221346706997,210723727773531,1264832205228154,7614504644529573,45965528482747194,278169935575048042,1687298108487873673,10256585388232231101,62470753620679133927,381201089984659788693,2330136670337522460729,14266260150998229954489

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

%H Paul D. Hanna, <a href="/A268039/b268039.txt">Table of n, a(n) for n = 1..500</a>

%F a(n) ~ c * d^n / n^(3/2), where d = 6.426300389361325672327464898259648... and c = 0.035189970759375828095135204598637... . - _Vaclav Kotesovec_, May 03 2016

%e G.f.: A(x) = x + x^2 + 2*x^3 + 9*x^4 + 40*x^5 + 192*x^6 + 959*x^7 + 4988*x^8 + 26587*x^9 + 144672*x^10 + 800229*x^11 + 4486914*x^12 +...

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

%e RELATED SERIES.

%e A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 40*x^6 + 192*x^7 + 963*x^8 + 4988*x^9 + 26589*x^10 + 144672*x^11 + 800253*x^12 + 4486914*x^13 + 25444374*x^14 +...

%e x*A(x) - A(x)^3 = x^2 - x^4 - 4*x^8 - 2*x^10 - 24*x^12 - 19*x^14 - 206*x^16 - 194*x^18 - 1980*x^20 - 2390*x^22 - 20920*x^24 - 31626*x^26 - 236114*x^28 +...

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

%e B(x) = x - x^2 - 4*x^4 - 2*x^5 - 24*x^6 - 19*x^7 - 206*x^8 - 194*x^9 - 1980*x^10 - 2390*x^11 - 20920*x^12 - 31626*x^13 - 236114*x^14 +...

%e such that B(x^2) = x*A(x) - A(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*F - F^3))[#A] ); A[n]}

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

%Y Cf. A265940.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Apr 27 2016