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A271957 G.f. A(x) satisfies: A(x) = A( x^2 + 10*x*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.

%I A271957 #8 Apr 19 2016 18:18:17
%S A271957 1,5,40,375,3845,41825,474450,5552250,66548785,812875800,10082125950,
%T A271957 126637168125,1607562407775,20591392666250,265810034489750,
%U A271957 3454516382881875,45162288467005155,593528625987396725,7836767285955169200,103908861022437312375,1382961699685548183750,18469547560714428659250,247433242662040209056250,3324296142183357299203125,44779542961314348791789400,604655933814703316140014375
%N A271957 G.f. A(x) satisfies: A(x) = A( x^2 + 10*x*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.
%C A271957 Compare the g.f. to the following related identities:
%C A271957 (1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2), where C(x) = x + C(x)^2 (A000108).
%C A271957 (2) F(x) = F( x^2 + 4*x*F(x)^2 )^(1/2), where F(x) = D(x)^2/x and D(x) = x + D(x)^3/x (A001764).
%H A271957 Paul D. Hanna, <a href="/A271957/b271957.txt">Table of n, a(n) for n = 1..200</a>
%F A271957 G.f. A(x) satisfies: A( x*G(x^2) - 5*x^2 ) = x, where G(x)^2 = G(x^2) + 20*x, and G(x) is the g.f. of A264414.
%e A271957 G..f.: A(x) = x + 5*x^2 + 40*x^3 + 375*x^4 + 3845*x^5 + 41825*x^6 + 474450*x^7 + 5552250*x^8 + 66548785*x^9 + 812875800*x^10 + 10082125950*x^11 + 126637168125*x^12 +...
%e A271957 where A(x)^2 = A( x^2 + 10*x*A(x)^2 ).
%e A271957 RELATED SERIES.
%e A271957 A(x)^2 = x^2 + 10*x^3 + 105*x^4 + 1150*x^5 + 13040*x^6 + 152100*x^7 + 1815375*x^8 + 22078750*x^9 + 272728845*x^10 + 3412891200*x^11 + 43178951325*x^12 +...
%e A271957 Let B(x) be the series reversion of the g.f. A(x), A(B(x)) = x, then:
%e A271957 B(x) = x - 5*x^2 + 10*x^3 - 45*x^5 + 450*x^7 - 5535*x^9 + 75600*x^11 - 1106100*x^13 + 16953750*x^15 +...+ A264414(n)*x^(2*n+1) +...
%e A271957 such that B(x) = x*G(x^2) - 5*x^2 where G(x)^2 = G(x^2) + 20*x, and G(x) is the g.f. of A264414.
%o A271957 (PARI) {a(n) = my(A=x+x^2,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, x^2 + 10*X*A^2)^(1/2) ); polcoeff(A,n)}
%o A271957 for(n=1,30,print1(a(n),", "))
%Y A271957 Cf. A264414, A271930, A271935, A271931, A271934.
%K A271957 nonn
%O A271957 1,2
%A A271957 _Paul D. Hanna_, Apr 17 2016