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A264413 G.f. A(x) satisfies: A(x)^2 = A(x^2) + 12*x.

%I A264413 #13 Apr 16 2016 23:15:10
%S A264413 1,6,-15,90,-660,5310,-45765,413640,-3864345,37014120,-361577790,
%T A264413 3588484140,-36079979085,366728363460,-3762120325140,38901621985290,
%U A264413 -405039437707575,4242802537386450,-44681704461745740,472795814216587140,-5024232597805717410,53596341229925979360,-573736849659978481665,6161218734911098973490,-66355728143871653462745
%N A264413 G.f. A(x) satisfies: A(x)^2 = A(x^2) + 12*x.
%H A264413 Paul D. Hanna, <a href="/A264413/b264413.txt">Table of n, a(n) for n = 0..300</a>
%F A264413 Given g.f. A(x), let G(x) denote the g.f. of A264225, then:
%F A264413 (1) G( x/(A(x)^2 - 9*x) ) = x,
%F A264413 (2) G( x/(A(x^2) + 3*x) ) = x,
%F A264413 (3) A(G(x))^2 = (1+9*x) * G(x)/x,
%F A264413 (4) A(G(x)^2) = (1-3*x) * G(x)/x,
%F A264413 where G(x)^2 = G( x^2/(1-6*x) ).
%e A264413 G.f.: A(x) = 1 + 6*x - 15*x^2 + 90*x^3 - 660*x^4 + 5310*x^5 - 45765*x^6 + 413640*x^7 - 3864345*x^8 + 37014120*x^9 - 361577790*x^10 +...
%e A264413 where
%e A264413 A(x)^2 = 1 + 12*x + 6*x^2 - 15*x^4 + 90*x^6 - 660*x^8 + 5310*x^10 - 45765*x^12 + 413640*x^14 - 3864345*x^16 + 37014120*x^18 - 361577790*x^20 +...
%e A264413 so that A(x)^2 = A(x^2) + 12*x.
%e A264413 Let G(x) = Series_Reversion( x / (A(x^2) + 3*x) ), then
%e A264413 G(x) = x + 3*x^2 + 15*x^3 + 81*x^4 + 462*x^5 + 2718*x^6 + 16344*x^7 + 99900*x^8 + 618567*x^9 + 3870909*x^10 +...+ A264225(n)*x^n +...
%e A264413 such that G(x)^2 = G( x^2/(1-6*x) ).
%o A264413 (PARI) {a(n) = my(A=1); for(i=1,n, A = sqrt( subst(A,x,x^2) + 12*x +x*O(x^n))); polcoeff(A,n)}
%o A264413 for(n=0,30,print1(a(n),", "))
%Y A264413 Cf. A271935, A264225, A264233, A264412, A264414, A264415.
%K A264413 sign
%O A264413 0,2
%A A264413 _Paul D. Hanna_, Nov 12 2015