A158093 a(n) = 3^(n^2+n)*C(1/3^n, n) = [x^n] (1 + 3^(n+1)*x)^(1/3^n).

1, 3, -36, 6201, -10519740, 168009075234, -24937507748845692, 34147337933260567913832, -429040882807948915054596365580, 49262806958277650055073574841789707655

Offset

0,2

Comments

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..45

Formula

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!.

Example

G.f.: A(x) = 1 +3*x/3^2 -36*x^2/3^6 +6201*x^3/3^12 -10519740*x^4/3^20 +...
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! +...
Illustrate a(n) = [x^n] (1 + 3^(n+1)*x)^(1/3^n):
(1+9*x)^(1/3) = 1 + (3)*x - 9*x^2 + 45*x^3 - 270*x^4 +...
(1+27*x)^(1/9) = 1 + 3*x - (36)*x^2 + 612*x^3 - 11934*x^4 +...
(1+81*x)^(1/27) = 1 + 3*x - 117*x^2 + (6201)*x^3 - 372060*x^4 +...
(1+243*x)^(1/81) = 1 + 3*x - 360*x^2 + 57960*x^3 - (10519740)*x^4 +...
Special values of A(x).
A(1) = 1 + log(4/3) + log(10/9)^2/2! + log(28/27)^3/3! +...
A(3) = 1 + log(2) + log(4/3)^2/2! + log(10/9)^3/3! +...
A(9) = 1 + log(4) + log(2)^2/2! + log(4/3)^3/3! + log(10/9)^4/4! +...
A(r) = 2 at r=4.50548200106313905...
A(r) = 3 at r=12.21509538023664538...
A(r) = 4 at r=22.9609516534592247304...

Prog

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

Crossrefs

Cf. A159478, A159558, A183131.

Sequence in context: A136393 A168370 A325907 * A163966 A262825 A088322

Adjacent sequences: A158090 A158091 A158092 * A158094 A158095 A158096

Keyword

sign

Author

Paul D. Hanna, Apr 21 2009

Status

approved