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A356780 Coefficients in the odd function A(x) such that: A(x) = A( x^2 + 2*x^2*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.

%I A356780 #11 Sep 02 2022 18:55:35
%S A356780 1,1,2,6,21,78,303,1223,5085,21623,93585,410894,1825682,8193544,
%T A356780 37087449,169114547,776110247,3581944258,16614576945,77410877233,
%U A356780 362126147797,1700179143293,8008689767674,37838553977426,179268540549758,851478474635404,4053760582437106
%N A356780 Coefficients in the odd function A(x) such that: A(x) = A( x^2 + 2*x^2*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.
%C A356780 Compare the g.f. to the following identities:
%C A356780 (1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),
%C A356780 (2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
%C A356780 where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
%F A356780 G.f. A(x) = Sum_{n>=1} a(n) * x^(2*n-1) satisfies:
%F A356780 (1) A(x) = sqrt( A( x^2 + 2*x^2*A(x)^2 ) ).
%F A356780 (2) G(x) = sqrt( G(x^2) / (1 + 2*x^2) ), where A(G(x)) = x.
%e A356780 G.f. A(x) = x + x^3 + 2*x^5 + 6*x^7 + 21*x^9 + 78*x^11 + 303*x^13 + 1223*x^15 + 5085*x^17 + 21623*x^19 + 93585*x^21 + ...
%e A356780 where A(x)^2 = A( x^2 + 2*x^2*A(x)^2 ).
%e A356780 RELATED SERIES.
%e A356780 A(x)^2 = x^2 + 2*x^4 + 5*x^6 + 16*x^8 + 58*x^10 + 222*x^12 + 882*x^14 + 3616*x^16 + 15205*x^18 + 65220*x^20 + ...
%e A356780 x^2 + 2*x^2*A(x)^2 = x^2 + 2*x^4 + 4*x^6 + 10*x^8 + 32*x^10 + 116*x^12 + 444*x^14 + 1764*x^16 + 7232*x^18 + 30410*x^20 + ...
%e A356780 Let G(x) = Series_Reversion( A(x) ) then
%e A356780 G(x) = x - x^3 + x^5 - 2*x^7 + 4*x^9 - 7*x^11 + 12*x^13 - 23*x^15 + 45*x^17 - 84*x^19 + 157*x^21 - 302*x^23 + 584*x^25 - 1121*x^27 + ...
%e A356780 where G(x)^2 = G(x^2)/(1 + 2*x^2) and G(A(x)) = x.
%o A356780 (PARI) {a(n) = my(A=x+x^3, X=x+x*O(x^(2*n))); for(i=1, 2*n, A = subst(A, x, x^2 + 2*X^2*A^2)^(1/2) ); polcoeff(A, 2*n-1)}
%o A356780 for(n=1, 30, print1(a(n), ", "))
%Y A356780 Cf. A000108, A356781, A271931, A271932, A271933.
%K A356780 nonn
%O A356780 1,3
%A A356780 _Paul D. Hanna_, Aug 27 2022