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%I #24 Jul 29 2022 21:27:11
%S 1,3,9,36,135,567,2268,9720,40095,173745,729729,3184272,13533156,
%T 59337684,254304360,1118939184,4825425231,21288640725,92250776475,
%U 407845538100,1774128090735,7856852973255,34284449337840,152044079672160,665192848565700,2953456247631708,12949769701154412,57554532005130720,252828837022538520,1124652412962326520,4948470617034236688,22028675650023376224
%N G.f. A(x) satisfies: A(x - 3*x^3) = 1/(1 - 3*x).
%C Compare with the o.g.f. G(x) = (1 - sqrt(1 - 8*x))/(4*x) of A151374, which satisfies G(x - 2*x^2) = 1/(1 - 2*x). - _Peter Bala_, Oct 29 2017
%D R. L. Graham, D. E. Knuth & O. Patashnik, Concrete Mathematics (1989), Addison-Wesley, Reading, MA. Sections 5.4 & 7.5.
%H N. D. Elkies, <a href="http://www.math.harvard.edu/~elkies/Misc/catalan.pdf">The power series for the inverse function of y(1-y)^t</a>
%F G.f. A(x) satisfies:
%F (1) A(x) * A(-x) = (A(x) + A(-x))/2.
%F (2) A(x) = 1/(1 - F(x)^2) where F(x) = sqrt(3) * G(sqrt(3)*x) where G(x) = x + G(x)^3 is the g.f. of A001764.
%F (3) (A(x) - A(-x)) / (A(x) + A(-x)) = sqrt(3) * G(sqrt(3)*x) such that G(x) = x + G(x)^3 is the g.f. of A001764.
%F (4) A'(x) = 3*A(x)^3 * A(-x).
%F (5) A(x) = exp( Integral 3*A(x)^2 * A(-x) dx ).
%F From _Peter Bala_, Oct 29 2017: (Start)
%F a(2*n) = 3^n*binomial(3*n, n).
%F a(2*n+1) = 3^(n+1)*binomial(3*n+1, n).
%F The proof uses the properties of Lambert's generalized binomial series B_t(z) at t = 3 - see Graham et al., sections 5.4 and 7.5 and also Elkies. (End)
%F D-finite with recurrence: 4*n*(n-1)*a(n) +18*(-n+1)*a(n-1) -9*(3*n-4)*(3*n-5)*a(n-2)=0. - _R. J. Mathar_, Jan 27 2020
%e G.f.: A(x) = 1 + 3*x + 9*x^2 + 36*x^3 + 135*x^4 + 567*x^5 + 2268*x^6 + 9720*x^7 + 40095*x^8 + 173745*x^9 +...
%e such that A(x - 3*x^3) = 1/(1 - 3*x).
%e RELATED SERIES.
%e A(x) * A(-x) = 1 + 9*x^2 + 135*x^4 + 2268*x^6 + 40095*x^8 + 729729*x^10 + 13533156*x^12 + 254304360*x^14 + 4825425231*x^16 +...
%e which equals (A(x) + A(-x))/2.
%e (A(x) - A(-x))/2 = 3*x + 36*x^3 + 567*x^5 + 9720*x^7 + 173745*x^9 + 3184272*x^11 + 59337684*x^13 + 1118939184*x^15 + 21288640725*x^17 +...
%e (A(x) - A(-x)) / (A(x) + A(-x)) = sqrt(3) * G(sqrt(3)*x) where
%e G(x) = x + x^3 + 3*x^5 + 12*x^7 + 55*x^9 + 273*x^11 + 1428*x^13 + 7752*x^15 +...+ binomial(3*n,n)/(2*n+1)*x^(2*n+1) +...
%e such that G(x) = x + G(x)^3 is the g.f. of A001764.
%t nmax = 31; sol = {a[0] -> 1};
%t Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x - 3 x^3] - 1/(1 - 3 x) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
%t sol /. Rule -> Set;
%t a /@ Range[0, nmax] (* _Jean-François Alcover_, Nov 02 2019 *)
%o (PARI) {a(n) = my(A=1); A = 1/(1 - 3*serreverse(x - 3*x^3 +x^2*O(x^n))); polcoeff(A,n)}
%o for(n=0,35,print1(a(n),", "))
%o (PARI) {a(n) = my(A=1); for(i=1,n, A = exp(intformal( 3*A^2*subst(A,x,-x) +x*O(x^n)))); polcoeff(A,n)}
%o for(n=0,35,print1(a(n),", "))
%Y Cf. A001764, A151374.
%K nonn,easy
%O 0,2
%A _Paul D. Hanna_, Sep 02 2016
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