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%I #6 Sep 08 2022 08:45:43
%S 1,3,126,66708,379033074,21399656315607,11566324342205917416,
%T 58678275719834357303044728,2762222169999029718435709903699050,
%U 1197781369953334546750963984948238943438411
%N a(n) = 3^(n^2+n) * C(2*n-1 + 1/3^n, n) / (n*3^n + 1).
%H G. C. Greubel, <a href="/A159319/b159319.txt">Table of n, a(n) for n = 0..45</a>
%F G.f.: A(x) = Sum_{n>=0} a(n)*x^n/3^(n^2+n).
%F G.f.: A(x) = Sum_{n>=0} log(F(x/3^n))^n/n! and
%F a(n)/3^(n^2+n) = [x^n] F(x)^(1/3^n) where
%F F(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108).
%F a(n)/3^(n^2+n) = [x^n] 1/(1-x)^(n + 1/3^n)/(n*3^n + 1).
%F Radius of convergence of series A(x) is |x| <= 3/4.
%e G.f.: A(x) = 1 + 3*x/3^2 + 126*x^2/3^6 + 66708*x^3/3^12 + 379033074*x^4/3^20 +...
%e A(x) = Sum_{n>=0} log( (1-sqrt(1-4*x/3^n))/(2*x/3^n) )^n/n!.
%e A(x) = 1 + log(F(x/3)) + log(F(x/9))^2/2! + log(F(x/27))^3/3! +... where F(x) = (1-sqrt(1-4*x))/(2*x).
%e Special values.
%e A(3/4) = 1 + log(2) + log(6-6*sqrt(2/3))^2/2! + log(18-18*sqrt(8/9))^3/3! + log(54-54*sqrt(26/27))^4/4! +...
%e A(3/4) = 1.6977820781412737038286578011417848301231627494589650...
%e A(-3/4) = 1 + log(2*sqrt(2)-2) + log(6*sqrt(4/3)-6)^2/2! + log(18*sqrt(10/9)-18)^3/3! + log(54*sqrt(28/27)-54)^4/4! +...
%e A(-3/4) = 0.8145458917316632938137444904602229430460096517471900...
%e Illustrate (3^n)-th root formula:
%e a(n)/3^(n^2+n) = [x^n] F(x)^(1/3^n) or, equivalently,
%e a(n) = [x^n] F(3^(n+1)*x)^(1/3^n) where F(x)=Catalan(x):
%e F(3*x) = (1) + 3*x + 18*x^2 + 135*x^3 + 1134*x^4 + 10206*x^5 +...
%e F(9*x)^(1/3) = 1 + (3)*x + 45*x^2 + 936*x^3 + 22572*x^4 +...
%e F(27*x)^(1/9) = 1 + 3*x + (126)*x^2 + 7659*x^3 + 546480*x^4 +...
%e F(81*x)^(1/27) = 1 + 3*x + 369*x^2 + (66708)*x^3 + 14215230*x^4 +...
%e F(243*x)^(1/81) = 1 + 3*x + 1098*x^2 + 593775*x^3 + (379033074)*x^4 +...
%e coefficients in parenthesis form the initial terms of this sequence.
%t Table[3^(n^2 +n)*Binomial[2*n -1 +1/3^n, n]/(n*3^n +1), {n, 0, 50}] (* _G. C. Greubel_, Jun 26 2018 *)
%o (PARI) {a(n)=3^(n^2+n)*binomial(2*n-1+1/3^n, n)/(n*3^n + 1)}
%o (PARI) {a(n)=3^(n^2+n)*polcoeff(1/(1-x+x*O(x^n))^(n+1/3^n)/(n*3^n + 1),n)}
%o (PARI) {a(n)=3^(n^2+n)*polcoeff(((1-sqrt(1-4*x+x^2*O(x^n)))/(2*x))^(1/3^n),n)}
%o (PARI) {a(n)=3^(n^2+n)*polcoeff(sum(k=0,n,log((1-sqrt(1-4*x/3^k+x^2*O(x^n)))/(2*x/3^k))^k/k!),n)}
%o (Magma) [3^(n^2 +n)*Binomial(2*n -1 +1/3^n, n)/(n*3^n +1): n in [0..40]]; // _G. C. Greubel_, Jun 26 2018
%Y Cf. A159318, A158093, A159558, A159478, A000108.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Apr 23 2009
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