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%I #10 Jan 29 2022 02:58:57
%S 1,3,45,6930,11006901,170914738743,25094213868636459,
%T 34223931232325024404500,429380124224154112727394224229,
%U 49276413396437070036301200874674619650,51540221234153816900126405027724655190954838780
%N a(n) = 3^(n^2+n) * C(n-1 + 1/3^n, n).
%H Seiichi Manyama, <a href="/A183131/b183131.txt">Table of n, a(n) for n = 0..45</a>
%F a(n) = [x^n] 1/(1 - 3^(n+1)*x)^(1/3^n).
%F G.f.: Sum_{n>=0} (-log(1 - x/3^n))^n/n! = Sum_{n>=0} a(n)*x^n/3^(n^2+n).
%e G.f.: A(x) = 1 + 3*x/3^2 + 45*x^2/3^6 + 6930*x^3/3^12 + 11006901*x^4/3^20 + 170914738743*x^5/3^30 +...
%e A(x) = 1 - log(1-x/3) + log(1-x/9)^2/2! - log(1-x/27)^3/3! + log(1-x/81)^4/4! +...+ (-1)^n*log(1-x/3^n)^n/n! +...
%e Illustrate a(n) = [x^n] 1/(1 - 3^(n+1)*x)^(1/3^n):
%e (1-9*x)^(-1/3) = 1 + (3)*x + 18*x^2 + 126*x^3 + 945*x^4 +...
%e (1-27*x)^(-1/9) = 1 + 3*x + (45)*x^2 + 855*x^3 + 17955*x^4 +...
%e (1-81*x)^(-1/27) = 1 + 3*x + 126*x^2 + (6930)*x^3 + 426195*x^4 +...
%e (1-243*x)^(-1/81) = 1 + 3*x + 369*x^2 + 60147*x^3 + (11006901)*x^4 +...
%e (1-729*x)^(-1/243) = 1 + 3*x + 1098*x^2 + 534726*x^3 + 292762485*x^4 + (170914738743)*x^5 +...
%e Special values.
%e A(1) = 1 + log(3/2) + log(9/8)^2/2! + log(27/26)^3/3! + log(81/80)^4/4! +...
%e A(-1) = 1 + log(3/4) + log(9/10)^2/2! + log(27/28)^3/3! + log(81/82)^4/4! +...
%e A(1/3) = 1 + log(9/8) + log(27/26)^2/2! + log(81/80)^3/3! + log(243/242)^4/4! +...
%e A(4/3) = 1 + log(9/5) + log(27/23)^2/2! + log(81/77)^3/3! + log(243/239)^4/4! +...
%e A(1) = 1.412410489973035808125672257400880...
%e A(-1) = 0.7178603309478784469203322438498398552...
%e A(-3) = 0.3480384480558263511525077084408616142...
%e A(3) is indeterminate.
%o (PARI) a(n)=3^(n^2+n)*binomial(n-1+1/3^n, n)
%o (PARI) {a(n)=3^(n^2+n)*polcoeff(1+sum(m=1,n,(-log(1 - x/3^m +x*O(x^n)))^m/m!),n)}
%Y Cf. A159558, A158093.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 26 2010
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