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A357548 a(n) = coefficient of x^n in A(x) where A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ).

%I A357548 #19 Dec 04 2022 07:36:01
%S A357548 1,2,11,50,261,1362,7344,40112,222338,1245476,7043605,40153390,
%T A357548 230518723,1331576430,7733934030,45138530004,264596552838,
%U A357548 1557101158092,9195520745412,54477134410680,323668083179382,1928047124332764,11512382184408072,68889282756213840
%N A357548 a(n) = coefficient of x^n in A(x) where A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ).
%C A357548 Radius of convergence is r = (sqrt(57) - 5)/16, where r = r^2/(1 - 4*r - 8*r^2), with A(r) = 1.
%C A357548 Related identities:
%C A357548 (1) F(x)^2 = F( x^2/(1 - 4*x + 6*x^2) ) when F(x) = x/(1-2*x).
%C A357548 (2) C(x)^2 = C( x^2/(1 - 4*x + 4*x^2) ) when C(x) = (1-2*x - sqrt(1-4*x))/(2*x) is a g.f. of the Catalan numbers (A000108).
%C A357548 More generally, if
%C A357548 F(x)^2 = F( x^2/(1 - 2*a*x + 2*(a^2 - b)*x^2) ),
%C A357548 then
%C A357548 F( x/(1 + a*x + b*x^2) )^2 = F( x^2/(1 + a^2*x^2 + b^2*x^4) );
%C A357548 here, a = 2, b = 8.
%H A357548 Paul D. Hanna, <a href="/A357548/b357548.txt">Table of n, a(n) for n = 1..520</a>
%F A357548 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
%F A357548 (1) A( x/(1 + 2*x + 8*x^2) )^2 = A( x^2/(1 + 2^2*x^2 + 8^2*x^4) ).
%F A357548 (2) A(x) = -A( -x/(1 - 4*x) ).
%F A357548 (3) A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ).
%F A357548 (4) A( x/(1 + 2*x) )^2 = A( x^2/(1 - 12*x^2) ).
%F A357548 (5) A( x/(1 + 4*x) )^2 = A( x^2/(1 + 4*x - 8*x^2) ).
%e A357548 G.f.: A(x) = x + 2*x^2 + 11*x^3 + 50*x^4 + 261*x^5 + 1362*x^6 + 7344*x^7 + 40112*x^8 + 222338*x^9 + 1245476*x^10 + 7043605*x^11 + 40153390*x^12 + ...
%e A357548 where A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ).
%e A357548 RELATED SERIES.
%e A357548 A(x)^2 = x^2 + 4*x^3 + 26*x^4 + 144*x^5 + 843*x^6 + 4868*x^7 + 28378*x^8 + 165664*x^9 + 971013*x^10 + 5708132*x^11 + 33660362*x^12 + ...
%e A357548 (x*A(x))^(1/2) = x + x^2 + 5*x^3 + 20*x^4 + 98*x^5 + 483*x^6 + 2499*x^7 + 13182*x^8 + 71030*x^9 + 388484*x^10 + ... + A357786(n)*x^n + ...
%e A357548 x/Series_Reversion(A(x)) = 1 + 2*x + 7*x^2 - 21*x^4 + 147*x^6 - 1260*x^8 + 11907*x^10 - 120540*x^12 + 1279047*x^14 - 14029428*x^16 + 157788183*x^18 + ...
%o A357548 (PARI) {a(n) = my(A=x); for(i=1, #binary(n+1),
%o A357548 A = sqrt( subst(A, x, x^2/(1 - 4*x - 8*x^2 +x*O(x^n)) ) )
%o A357548 ); polcoeff(A, n)}
%o A357548 for(n=1, 40, print1(a(n), ", "))
%Y A357548 Cf. A357786, A264224, A274483, A274484, A357547.
%K A357548 nonn
%O A357548 1,2
%A A357548 _Paul D. Hanna_, Dec 01 2022