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A371708 Expansion of g.f. A(x) satisfying A( x*A(x - x^2) ) = x^2.

%I A371708 #25 May 05 2024 20:11:35
%S A371708 1,1,1,2,6,19,60,193,636,2141,7331,25451,89385,317036,1134100,4087104,
%T A371708 14825482,54088470,198348985,730723956,2703194553,10037648254,
%U A371708 37399878530,139785998185,523962161491,1969154471389,7418488063284,28010998254007,105986233046356,401804972780552
%N A371708 Expansion of g.f. A(x) satisfying A( x*A(x - x^2) ) = x^2.
%H A371708 Paul D. Hanna, <a href="/A371708/b371708.txt">Table of n, a(n) for n = 1..520</a>
%F A371708 G.f. A(x) = Sum_{n>=1} a(n)*x^n, along with its series reversion R(x), satisfy the following formulas.
%F A371708 (1) A( x*A(x - x^2) ) = x^2.
%F A371708 (2) A(x - x^2) = R(x^2)/x.
%F A371708 (3) (R(x) - R(-x))^2 + 2*(R(x) + R(-x)) = 0.
%F A371708 (4) R(x) = R(-x) - 1 + sqrt(1 - 4*R(-x)).
%F A371708 (5) A(x) = -A( x - 1 + sqrt(1 - 4*x) ).
%F A371708 (6) A(x) = -A(x - 2*C(x)) where C(x) = -C(x - 2*C(x)) is a g.f. of the Catalan numbers (A000108).
%F A371708 (7) A( A(x)*C(x) ) = C(x)^2 where C(x) = (1 - sqrt(1 - 4*x))/2 is a g.f. of the Catalan numbers (A000108).
%F A371708 a(n) ~ c * 4^n / n^(3/2), where c = 0.0517683007874758928168667... - _Vaclav Kotesovec_, Apr 24 2024
%e A371708 G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 6*x^5 + 19*x^6 + 60*x^7 + 193*x^8 + 636*x^9 + 2141*x^10 + 7331*x^11 + 25451*x^12 + 89385*x^13 + 317036*x^14 + ...
%e A371708 where A( x*A(x - x^2) ) = x^2.
%e A371708 RELATED SERIES.
%e A371708 Let R(x) be the series reversion of A(x), A(R(x)) = x, which begins
%e A371708 R(x) = x - x^2 + x^3 - 2*x^4 + 2*x^5 - 5*x^6 + 6*x^7 - 16*x^8 + 23*x^9 - 62*x^10 + 100*x^11 - 270*x^12 + 463*x^13 - 1254*x^14 + 2224*x^15 - 6050*x^16 + ...
%e A371708 then R( R(x^2)/x ) = x - x^2.
%e A371708 Also, the bisections B1 and B2 of R(x) are
%e A371708 B1 = (R(x) - R(-x))/2 = x + x^3 + 2*x^5 + 6*x^7 + 23*x^9 + 100*x^11 + 463*x^13 + 2224*x^15 + 10963*x^17 + ...
%e A371708 B2 = (R(x) + R(-x))/2 = -x^2 - 2*x^4 - 5*x^6 - 16*x^8 - 62*x^10 - 270*x^12 - 1254*x^14 - 6050*x^16 + ...
%e A371708 and satisfy B1^2 + B2 = 0 and A(x*B1) = B1^2.
%e A371708 SPECIFIC VALUES.
%e A371708 A( A(1/4) / 2 ) = 1/4 where
%e A371708 A(1/4) = 0.39241307250698647662923990494867613212061604622566765...
%e A371708 A( A(2/9) / 3 ) = 1/9 where
%e A371708 A(2/9) = 0.29957319341272312632777466712131772539171747971866175...
%e A371708 A( A(3/16) / 4 ) = 1/16 where
%e A371708 A(3/16) = 0.2352360051274118086289466324430753987734355106832392...
%e A371708 A( A(4/25) / 5 ) = 1/25 where
%e A371708 A(4/25) = 0.1922953260179964363449115205476634347705922222443464...
%o A371708 (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
%o A371708 A[#A] = polcoeff( x^2 - subst(Ser(A),x, x*subst(Ser(A),x, x - x^2) ), #A));A[n+1]}
%o A371708 for(n=1,35,print1(a(n),", "))
%Y A371708 Cf. A372528, A272483, A213591, A000108.
%K A371708 nonn
%O A371708 1,4
%A A371708 _Paul D. Hanna_, Apr 23 2024