A272463 - OEIS

%I #14 Apr 08 2024 04:57:37

%S 1,1,2,4,12,36,112,360,1184,3969,13506,46550,162160,570076,2019864,

%T 7205654,25859788,93299268,338207096,1231194329,4499137382,

%U 16498152995,60689045230,223891151262,828156420320,3070760666368,11411884518800,42498639965025,158575098942194,592761262161240,2219513277432130,8323806778346002,31262959171040784,117582688976071889,442819400938052362,1669735077567533522,6303424784771599874,23822450122837267190

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

%H Paul D. Hanna, <a href="/A272463/b272463.txt">Table of n, a(n) for n = 1..500</a>

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

%F (1) B(x)^2 = B(B(x)^3) / (x - x^2).

%F (2) B(x^3)/x^2 = A(x) - A(x)^2.

%F (3) A(x) = (1 - sqrt(1 - 4*B(x^3)/x^2)) / 2.

%F (4) A(x) = C( B(x^3)/x^2 ), where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers A000108.

%F a(n) ~ c * d^n / n^(3/2), where d = 3.9343346144956442477821996521921... and c = 0.1374726251259662065926106275441... . - _Vaclav Kotesovec_, May 03 2016

%F Let r be the radius of convergence, then A(r) = 1/2, where r = A(r^2/4)^(1/3) = 0.25417258519791494791315235901614229902947074103519177... = 1/d (d is given above). - _Paul D. Hanna_, Apr 06 2024

%e G.f.: A(x) = x + x^2 + 2*x^3 + 4*x^4 + 12*x^5 + 36*x^6 + 112*x^7 + 360*x^8 + 1184*x^9 + 3969*x^10 + 13506*x^11 + 46550*x^12 +...

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

%e RELATED SERIES.

%e A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 12*x^5 + 36*x^6 + 112*x^7 + 360*x^8 + 1184*x^9 + 3968*x^10 + 13506*x^11 + 46550*x^12 +...

%e A(x) - A(x)^2 = x - x^4 + x^10 - 4*x^13 + 6*x^16 - 27*x^22 + 84*x^25 - 119*x^28 - 70*x^31 + 861*x^34 - 2362*x^37 + 2716*x^40 + 4848*x^43 - 31892*x^46 +...

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

%e B(x) = x - x^2 + x^4 - 4*x^5 + 6*x^6 - 27*x^8 + 84*x^9 - 119*x^10 - 70*x^11 + 861*x^12 - 2362*x^13 + 2716*x^14 + 4848*x^15 - 31892*x^16 +...

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

%e SPECIFIC VALUES.

%e A(1/4) = 0.43750716413214762438474169851025169044...

%e A(1/5) = 0.2728438844373996476937912739143254714680748999753267...

%e A(1/6) = 0.2099915138524924668889213019413855047403227975244006...

%e A(1/7) = 0.1720375832939219643807299314288451907008185008379640...

%e A(1/8) = 0.1461015121522386794470333784876359422356861236711932...

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

%o for(n=1,50,print1(a(n),", "))

%Y Cf. A265940, A268039, A292927, A292928, A371714.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Apr 29 2016