OFFSET
0,3
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 y=3*x.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..412
FORMULA
E.g.f.: A(x) = Sum_{n>=0} x^n/n! * Product_{k=1..n} (1 + 3*k*x).
a(n) ~ n! * n^(1/sqrt(3))*3^(n/2+1/2)/(2^(1+1/sqrt(3))*Gamma(1/sqrt(3))). - Vaclav Kotesovec, Jun 25 2013
EXAMPLE
E.g.f.: A(x) = 1 + x + 7*x^2/2! + 28*x^3/3! + 289*x^4/4! + 2131*x^5/5! +...
where A(x) satisfies:
A(x)^(3*x/(1+3*x)) = 1 + 3*x^2 + 9*x^4 + 27*x^6 + 81*x^8 + 243*x^10 +...
Also,
A(x) = 1 + x*(1+3*x) + x^2*(1+3*x)*(1+6*x)/2! + x^3*(1+3*x)*(1+6*x)*(1+9*x)/3! + x^4*(1+3*x)*(1+6*x)*(1+9*x)*(1+12*x)/4! +...
The logarithm begins:
log(A(x)) = x + 3*x^2 + 3*x^3/2 + 9*x^4/2 + 9*x^5/3 + 27*x^6/3 + 27*x^7/4 +...
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * 3^floor(k/2)/floor((k+1)/2) * a(n-k)/(n-k)!. - Seiichi Manyama, Apr 30 2022
MATHEMATICA
CoefficientList[Series[1/(1-3*x^2)^(1+1/(3*x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 25 2013 *)
PROG
(PARI) {a(n)=n!*polcoeff(1/(1 - 3*x^2 +x^2*O(x^n))^((1+3*x)/(3*x)), n)}
(PARI) {a(n)=n!*polcoeff(sum(m=0, n, x^m/m!*prod(k=1, m, 1+3*k*x+x*O(x^n))), n)}
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, j*3^(j\2)/((j+1)\2)*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Apr 30 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 07 2011
STATUS
approved