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A260650 G.f. satisfies: A(x)^2 = A( x^2/(1-4*x)^2 ).

%I A260650 #13 May 06 2024 13:11:33
%S A260650 1,4,18,88,455,2444,13486,75912,433935,2511388,14684422,86611848,
%T A260650 514704064,3078845696,18523994024,112026315616,680626958899,
%U A260650 4152411174284,25428402204982,156247439709832,963048223399984,5952595420121536,36887847899094888,229132114803540320,1426367728966653535,8897049258366111004
%N A260650 G.f. satisfies: A(x)^2 = A( x^2/(1-4*x)^2 ).
%C A260650 Radius of convergence is r = (9 - sqrt(17))/32 where r = r^2/(1-4*r)^2 with A(r) = 1.
%C A260650 Compare to: C(x)^2 = C( x^2/(1-2*x)^2 ) where C(x) = (1-2*x-sqrt(1-4*x))/(2*x) is a g.f. of the Catalan numbers, A000108.
%H A260650 Paul D. Hanna, <a href="/A260650/b260650.txt">Table of n, a(n) for n = 1..520</a>
%F A260650 G.f. satisfies:
%F A260650 (1) A(x) = -A( -x/(1-8*x) ).
%F A260650 (2) A(x^2) = A( x/(1+4*x) )^2 = A( -x/(1-4*x) )^2.
%F A260650 (3) A( x/(1+2*x)^2 ) = -A( -x/(1-2*x)^2 ), an odd function.
%F A260650 (4) A( x/(1+2*x)^2 )^2 = A( x^2/(1+4*x^2)^2 ), an even function.
%e A260650 G.f.: A(x) = x + 4*x^2 + 18*x^3 + 88*x^4 + 455*x^5 + 2444*x^6 + 13486*x^7 + 75912*x^8 + 433935*x^9 + 2511388*x^10 + 14684422*x^11 + 86611848*x^12 +...
%e A260650 where A( x^2/(1-4*x)^2 ) = A(x)^2,
%e A260650 A( x^2/(1-4*x)^2 ) = x^2 + 8*x^3 + 52*x^4 + 320*x^5 + 1938*x^6 + 11696*x^7 + 70648*x^8 + 427776*x^9 + 2597831*x^10 + 15824664*x^11 + 96687516*x^12 +...
%e A260650 Also, A( x/(1+4*x) ) = A(x^2)^(1/2),
%e A260650 A( x/(1+4*x) ) = x + 2*x^3 + 7*x^5 + 30*x^7 + 143*x^9 + 726*x^11 + 3840*x^13 + 20904*x^15 + 116275*x^17 + 657798*x^19 + 3772912*x^21 + 21890152*x^23 +...
%e A260650 Let B(x) = x/Series_Reversion( A(x) ), so that A(x) = x*B(A(x)), then
%e A260650 B(x) = 1 + 4*x + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 + 82*x^14 - 233*x^16 + 668*x^18 - 1949*x^20 + 5802*x^22 +...+ A107087(n)*x^(2*n) +...
%e A260650 such that B(x) = F(x^2) + 4*x = F(x)^2 where F(x) is the g.f. of A107087.
%o A260650 (PARI) {a(n) = my(A=x); for(i=1,#binary(n+1), A = sqrt( subst(A,x, x^2/(1-4*x +x*O(x^n))^2) ) ); polcoeff(A,n)}
%o A260650 for(n=1,30,print1(a(n),", "))
%Y A260650 Cf. A264232, A264224, A107087, A107088.
%K A260650 nonn
%O A260650 1,2
%A A260650 _Paul D. Hanna_, Nov 16 2015