A158093 - OEIS

%I #7 Jan 29 2022 02:58:14

%S 1,3,-36,6201,-10519740,168009075234,-24937507748845692,

%T 34147337933260567913832,-429040882807948915054596365580,

%U 49262806958277650055073574841789707655

%N a(n) = 3^(n^2+n)*C(1/3^n, n) = [x^n] (1 + 3^(n+1)*x)^(1/3^n).

%C A(1) = Sum_{n>=0} C(1/3^n,n) = Sum_{n>=0} log(1+1/3^n)^n/n! = 1.293240509200709604261070...

%H Seiichi Manyama, <a href="/A158093/b158093.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) = Sum_{n>=0} log(1+x/3^n)^n/n!.

%e G.f.: A(x) = 1 +3*x/3^2 -36*x^2/3^6 +6201*x^3/3^12 -10519740*x^4/3^20 +...

%e A(x) = 1 + log(1+x/3) + log(1+x/9)^2/2! + log(1+x/27)^3/3! +...+ log(1+x/3^n)^n/n! +...

%e Illustrate a(n) = [x^n] (1 + 3^(n+1)*x)^(1/3^n):

%e (1+9*x)^(1/3) = 1 + (3)*x - 9*x^2 + 45*x^3 - 270*x^4 +...

%e (1+27*x)^(1/9) = 1 + 3*x - (36)*x^2 + 612*x^3 - 11934*x^4 +...

%e (1+81*x)^(1/27) = 1 + 3*x - 117*x^2 + (6201)*x^3 - 372060*x^4 +...

%e (1+243*x)^(1/81) = 1 + 3*x - 360*x^2 + 57960*x^3 - (10519740)*x^4 +...

%e Special values of A(x).

%e A(1) = 1 + log(4/3) + log(10/9)^2/2! + log(28/27)^3/3! +...

%e A(3) = 1 + log(2) + log(4/3)^2/2! + log(10/9)^3/3! +...

%e A(9) = 1 + log(4) + log(2)^2/2! + log(4/3)^3/3! + log(10/9)^4/4! +...

%e A(r) = 2 at r=4.50548200106313905...

%e A(r) = 3 at r=12.21509538023664538...

%e A(r) = 4 at r=22.9609516534592247304...

%o (PARI) a(n)=3^(n^2+n)*binomial(1/3^n,n)

%Y Cf. A159478, A159558, A183131.

%K sign

%O 0,2

%A _Paul D. Hanna_, Apr 21 2009