A371714 - OEIS

%I #27 Apr 08 2024 04:56:35

%S 1,1,2,5,13,40,126,409,1360,4611,15878,55384,195282,694910,2492454,

%T 9001405,32704855,119462142,438441266,1616001547,5979144981,

%U 22199682130,82685478702,308864831632,1156806962608,4343254831180,16343719170558,61630500821158,232854921227616,881378279895534

%N Expansion of g.f. A(x) satisfying A( x^3*A(x) - x^3*A(x)^2 ) = x^4.

%H Paul D. Hanna, <a href="/A371714/b371714.txt">Table of n, a(n) for n = 1..520</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

%F (1) A( x^3*A(x)*(1 - A(x)) ) = x^4.

%F (2) A( -x^3*A(x)*(1 - A(x)) ) = (1 - sqrt(1 + 4*x^4 - 4*x^8))/2.

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

%F a(n) ~ c * d^n / n^(3/2), where d = 3.9841244648060905977016688650241255776651... and c = 0.13991881826475367145488117165180720475565183... - _Vaclav Kotesovec_, Apr 05 2024

%F Let r be the radius of convergence, then A(r) = 1/2, where r = A(r^3/4)^(1/4) = 0.2509961746510523531... = 1/d (d is given above). - _Paul D. Hanna_, Apr 06 2024

%e G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 13*x^5 + 40*x^6 + 126*x^7 + 409*x^8 + 1360*x^9 + 4611*x^10 + 15878*x^11 + 55384*x^12 + ...

%e where A( x^3*A(x)*(1 - A(x)) ) = x^4.

%e RELATED SERIES.

%e (1) Let R(x) be the series reversion of A(x), R(A(x)) = x, where

%e R(x) = x - x^2 + x^5 - 5*x^6 + 10*x^7 - 10*x^8 + 10*x^9 - 46*x^10 + 180*x^11 - 420*x^12 + 665*x^13 - 1085*x^14 + 3150*x^15 - 10190*x^16 + ...

%e upon comparing the expansion of R(x) to the series

%e A(x)*(1 - A(x)) = x - x^5 + x^17 - 5*x^21 + 10*x^25 - 10*x^29 + 10*x^33 - 46*x^37 + 180*x^41 - 420*x^45 + 665*x^49 - 1085*x^53 + ...

%e we see that A(x)*(1 - A(x)) = R(x^4)/x^3.

%e (2) Let B(x) be the even bisection of A(x),

%e B(x) = x^2 + 5*x^4 + 40*x^6 + 409*x^8 + 4611*x^10 + 55384*x^12 + ...,

%e then

%e B( x^3*A(x)*(1 - A(x)) ) = x^8 - 2*x^12 + 6*x^16 - 20*x^20 + 72*x^24 - 272*x^28 + 1064*x^32 + ... + (-1)^(n-1)*A071356(n-1)*x^(8*n) + ...

%e that is,

%e B( x^3*A(x)*(1 - A(x)) ) = (1 + 2*x^4 - sqrt(1 + 4*x^4 - 4*x^8))/4.

%e SPECIFIC VALUES.

%e Let r be the radius of convergence, then A(r) = 1/2, and

%e r = A(r^3/4)^(1/4) = 0.2509961746510523531562794924202947105158...

%e A(1/4) = 0.4687500009132742494908083392815082722109...

%e A(1/5) = 0.2756788017179389881387593924191299703438...

%e A(1/6) = 0.2111022081144963995053917910635203605728...

%e A(1/8) = 0.1464034536677082355575260818928469620931...

%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^4 - subst(F, x, x^3*F - x^3*F^2), #A+3) ); A[n]}

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

%Y Cf. A291614, A272463, A071356.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Apr 04 2024