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A356781 Expansion of g.f. A(x) satisfying A(x) = A( x^2 + 2*x^2*A(x) )^(1/2), with A(0)=0, A'(0)=1.

%I A356781 #27 Mar 14 2024 08:00:17
%S A356781 1,1,1,2,4,7,14,32,74,172,408,978,2349,5662,13737,33568,82596,204618,
%T A356781 510208,1279544,3224828,8162144,20735397,52848816,135088609,346214873,
%U A356781 889451320,2290164276,5908894762,15274778235,39555942836,102603159040,266545251022
%N A356781 Expansion of g.f. A(x) satisfying A(x) = A( x^2 + 2*x^2*A(x) )^(1/2), with A(0)=0, A'(0)=1.
%C A356781 Compare the g.f. to the following identities:
%C A356781 (1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),
%C A356781 (2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
%C A356781 where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
%H A356781 Paul D. Hanna, <a href="/A356781/b356781.txt">Table of n, a(n) for n = 1..960</a>
%F A356781 G.f. A(x) = Sum_{n>=1} a(n) * x^n satisfies the following formulas.
%F A356781 (1) A(x) = sqrt( A( x^2 + 2*x^2*A(x) ) ).
%F A356781 (2) G(x) = sqrt( G(x^2)/(1 + 2*x) ), where A(G(x)) = x.
%F A356781 (3) G(x) = x*F(x^2)*(1 - x*C(x^2)) = x*F(x)^2*(1 - 4*x)/(1 - x*C(x)) where G(A(x)) = x, F(x) is the g.f. of A370540, and C(x) = (1 - sqrt(1-4*x))/(2*x) is the Catalan function (A000108). - _Paul D. Hanna_, Mar 12 2024
%F A356781 a(n) ~ c * d^n / n^(3/2), where d = 2.721808159464577... and c = 0.218334153814... - _Vaclav Kotesovec_, Mar 14 2024
%e A356781 G.f. A(x) = x + x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 14*x^7 + 32*x^8 + 74*x^9 + 172*x^10 + 408*x^11 + 978*x^12 + 2349*x^13 + 5662*x^14 + ...
%e A356781 where A(x)^2 = A( x^2 + 2*x^2*A(x) ).
%e A356781 RELATED SERIES.
%e A356781 A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 13*x^6 + 26*x^7 + 54*x^8 + 122*x^9 + 284*x^10 + 668*x^11 + 1597*x^12 + 3864*x^13 + 9394*x^14 + ...
%e A356781 x^2 + 2*x^2*A(x) = x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 8*x^7 + 14*x^8 + 28*x^9 + 64*x^10 + 148*x^11 + 344*x^12 + 816*x^13 + 1956*x^14 + ...
%e A356781 Let G(x) = Series_Reversion( A(x) ) then
%e A356781 G(x) = x - x^2 + x^3 - 2*x^4 + 4*x^5 - 7*x^6 + 12*x^7 - 23*x^8 + 45*x^9 - 84*x^10 + 157*x^11 - 302*x^12 + 584*x^13 - 1121*x^14 + ...
%e A356781 where G(x)^2 = G(x^2)/(1 + 2*x) and G(A(x)) = x.
%e A356781 Also, the series bisections of G(x) = B1(x) - B2(x) begin
%e A356781 B1(x) = x + x^3 + 4*x^5 + 12*x^7 + 45*x^9 + 157*x^11 + 584*x^13 + 2155*x^15 + 8110*x^17 + ... + A370540(n)*x^(2*n+1) + ...
%e A356781 B2(x) = x^2 + 2*x^4 + 7*x^6 + 23*x^8 + 84*x^10 + 302*x^12 + 1121*x^14 + 4175*x^16 + 15739*x^18 + ...
%e A356781 where B2(x)/B1(x) = (1 - sqrt(1 - 4*x^2))/(2*x) = x*C(x^2) where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).
%e A356781 SPECIFIC VALUES.
%e A356781 A(sqrt(8)/8) = 0.655524504794958661...
%e A356781 A(1/3) = 0.562844396405786198664639714324421729...
%e A356781 A(1/4) = 0.343586921250887627089213663537057899...
%e A356781 A(1/5) = 0.253260720427093363870348121973670974...
%o A356781 (PARI) {a(n) = my(A=[1],G); for(i=1,n, A=concat(A,0); G = x*Ser(A); A = Vec((subst(G,x, x^2 + 2*x^2*G) +x^2*O(x^#A))^(1/2)); ); A[n]}
%o A356781 for(n=1, 40, print1(a(n), ", "))
%Y A356781 Cf. A370540, A000108.
%K A356781 nonn
%O A356781 1,4
%A A356781 _Paul D. Hanna_, Aug 27 2022