OFFSET
0,2
COMMENTS
More generally, we have the identity:
Sum_{n>=0} (x^n/n!)*Product_{k=1..n} (1+k*y) = 1/(1 - x*y)^(1 + 1/y); here x=2*x, y=3*x.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
E.g.f.: A(x) = Sum_{n>=0} 2^n*x^n/n! * Product_{k=1..n} (1 + 3*k*x).
a(n) ~ n! * 6^(n/2)*(n/2)^sqrt(2/3)/(2*Gamma(1+sqrt(2/3))). - Vaclav Kotesovec, Sep 22 2013
EXAMPLE
E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 116*x^3/3! + 1456*x^4/4! + 18272*x^5/5! +...
where A(x) satisfies:
A(x)^(3*x/(1+3*x)) = 1 + 6*x^2 + 36*x^4 + 216*x^6 +...+ 6^n*x^(2*n) +...
Also,
A(x) = 1 + 2*x*(1+3*x) + 4*x^2*(1+3*x)*(1+6*x)/2! + 8*x^3*(1+3*x)*(1+6*x)*(1+9*x)/3! + 16*x^4*(1+3*x)*(1+6*x)*(1+9*x)*(1+12*x)/4! +...
The logarithm begins:
log(A(x)) = 2*x + 6*x^2 + 2*6*x^3/2 + 6^2*x^4/2 + 2*6^2*x^5/3 + 6^3*x^6/3 + 2*6^3*x^7/4 + 6^4*x^8/4 + 2*6^4*x^9/5 + 6^5*x^10/5 +...
MATHEMATICA
CoefficientList[Series[1/(1-6*x^2)^(1+1/(3*x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 22 2013 *)
PROG
(PARI) {a(n)=n!*polcoeff(1/(1 - 6*x^2 +x^2*O(x^n))^((1+3*x)/(3*x)), n)}
(PARI) {a(n)=n!*polcoeff(sum(m=0, n, 2^m*x^m/m!*prod(k=1, m, 1+3*k*x+x*O(x^n))), n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 07 2011
STATUS
approved