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A370540 Expansion of g.f. A(x) satisfying A(x)^2 = A(x^2) * (1 - x*C(x)) * (1 - x*C(x^2)) / (1 - 4*x) where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).

%I A370540 #16 Mar 14 2024 07:37:50
%S A370540 1,1,4,12,45,157,584,2155,8110,30587,116326,443984,1702272,6546563,
%T A370540 25252094,97638658,378351696,1468876958,5712276601,22247635905,
%U A370540 86765271643,338795469496,1324374411164,5182303804184,20297243177269,79564763550396,312137086267106,1225421059470049
%N A370540 Expansion of g.f. A(x) satisfying A(x)^2 = A(x^2) * (1 - x*C(x)) * (1 - x*C(x^2)) / (1 - 4*x) where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).
%H A370540 Paul D. Hanna, <a href="/A370540/b370540.txt">Table of n, a(n) for n = 0..600</a>
%F A370540 G.f. A(x) = Sum_{n>=1} a(n)*x^n and C(x) = (1 - sqrt(1-4*x))/2 satisfy the following formulas.
%F A370540 (1) A(x)^2 = A(x^2) * F(x) where F(x) = (1 - x*C(x)) * (1 - x*C(x^2)) / (1 - 4*x) is the g.f. of A370539.
%F A370540 (2) G( x*A(x^2)*(1 - x*C(x^2)) ) = x, where G(x) = G( x^2 + 2*x^2*G(x) )^(1/2) is the g.f. of A356781.
%F A370540 a(n) ~ c * 4^n / sqrt(n), where c = 0.3550434768046000612979284344613941075803... - _Vaclav Kotesovec_, Mar 14 2024
%e A370540 G.f.: A(x) = 1 + x + 4*x^2 + 12*x^3 + 45*x^4 + 157*x^5 + 584*x^6 + 2155*x^7 + 8110*x^8 + 30587*x^9 + 116326*x^10 + 443984*x^11 + ...
%e A370540 RELATED SERIES.
%e A370540 We may illustrate the formulas using the following related series expansions.
%e A370540 Recall that the Catalan function C(x) = (1 - sqrt(1-4*x))/2 begins
%e A370540 C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + ... + A000108(n)*x^n + ...
%e A370540 (1) By definition, A(x) = sqrt( A(x^2) * F(x) ) where
%e A370540 F(x) = (1 - x*C(x)) * (1 - x*C(x^2)) / (1 - 4*x) begins
%e A370540 F(x) = 1 + 2*x + 8*x^2 + 30*x^3 + 118*x^4 + 462*x^5 + 1824*x^6 + 7208*x^7 + 28558*x^8 + ... + A370539(n)*x^n + ...
%e A370540 (2) Also, G(x) = G( x^2 + 2*x^2*G(x) )^(1/2) begins
%e A370540 G(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 + ... + A356781(n)*x^n + ...
%e A370540 such that the series reversion of G(x) equals
%e A370540 x*A(x^2)*(1 - x*C(x^2)) = 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 + ...
%o A370540 (PARI) {a(n) = my(x = 'x + O('x^(n+4)), C(x) = (1 - sqrt(1 - 4*x))/(2*x), A = 1+x); for(i=1,n, A = sqrt( subst(A,'x,x^2) * (1 - x*C(x)) * (1 - x*C(x^2)) / (1 - 4*x) ) ); polcoeff(A,n);}
%o A370540 for(n=0,30, print1(a(n),", "))
%Y A370540 Cf. A370539, A356781, A000108.
%K A370540 nonn
%O A370540 0,3
%A A370540 _Paul D. Hanna_, Mar 12 2024