A228862 - OEIS

%I #13 Sep 03 2017 05:20:40

%S 1,1,1,2,5,11,24,58,146,365,922,2383,6243,16463,43748,117224,316157,

%T 857088,2334700,6388017,17546354,48361208,133710567,370744754,

%U 1030649811,2871950293,8020308614,22443012438,62919001546,176699520967,497039125163,1400236234543,3950262035542

%N G.f. satisfies: x = A(x - A(x^2 - A(x^3 - A(x^4 - A(x^5 -...))))).

%C The g.f. of A228863 equals the series reversion of the g.f. of this sequence.

%H Paul D. Hanna, <a href="/A228862/b228862.txt">Table of n, a(n) for n = 1..300</a>

%F a(n) ~ c * d^n / n^(3/2), where d = 2.9489373... and c = 0.27314... - _Vaclav Kotesovec_, Sep 03 2017

%e G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 5*x^5 + 11*x^6 + 24*x^7 + 58*x^8 +...

%e Let G(x) be the series reversion of A(x) (cf. A228863), then

%e (1) G(x) = x - x^2 + x^3 - 2*x^4 + 3*x^5 - 4*x^6 + 6*x^7 - 10*x^8 + 18*x^9 - 35*x^10 + 71*x^11 - 147*x^12 + 303*x^13 - 616*x^14 + 1244*x^15 +...

%e (2) G(x - G(x)) = x^2 - x^3 + x^4 - x^5 + x^7 - x^8 + x^10 - x^11 + 3*x^13 - 10*x^14 + 17*x^15 - 14*x^16 - 6*x^17 + 38*x^18 +...

%e (3) G(x^2 - G(x - G(x))) = x^3 - x^4 + x^5 - x^6 + x^7 - 2*x^8 + 3*x^9 - 3*x^10 + 3*x^11 - 5*x^12 + 8*x^13 - 9*x^14 + 10*x^15 +...

%e (4) G(x^3 - G(x^2 - G(x - G(x)))) = x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^11 - x^12 + x^13 - 2*x^14 + 2*x^15 - x^17 + x^18 - 3*x^19 + 4*x^20 +...

%e (5) G(x^4 - G(x^3 - G(x^2 - G(x - G(x))))) = x^5 - x^6 + x^7 - x^8 + x^9 - x^10 + x^11 - 2*x^12 + 3*x^13 - 3*x^14 + 3*x^15 - 4*x^16 + 5*x^17 +...

%e ...

%o (PARI) {a(n)=local(A=x+x^2,G=x^(n+1));for(i=1,n+1,A=serreverse(x-G+x^2*O(x^n));G=x^(n+1);for(k=0,n-1,G=subst(A,x,x^(n-k+1)-G+x^2*O(x^n))));polcoeff(A,n)}

%o for(n=1,35,print1(a(n),", "))

%Y Cf. A228863, A228835, A228883.

%K nonn

%O 1,4

%A _Paul D. Hanna_, Sep 05 2013