OFFSET
0,2
COMMENTS
Compare the g.f. to the LambertW identity:
1 = Sum_{n>=0} 2*(n+2)^(n-1) * exp(-(n+2)*x) * x^n/n!.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..345
FORMULA
a(n) = 1/n! * Sum_{k=0..n} 2*(-1)^(n-k)*binomial(n,k) * k^n * (k+2)^(n-1).
a(n) = 1/n! * [x^n] Sum_{k>=0} 2*k^k*(k+2)^(k-1)*x^k / (1 + k*(k+2)*x)^(k+1).
a(n) = [x^n] 1 + 2*x*(1+2*x)^(n-1) / Product_{k=1..n} (1-k*x).
a(n) = [x^n] 1 + 2*x*(1-2*x)^(n-1) / Product_{k=1..n} (1-(k+2)*x).
a(n) ~ 2^(2*n+1/2) * n^(n-3/2) / (sqrt(Pi*(1-c)) * exp(n) * (2-c)^(n-1) * c^(n+1)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 22 2014
EXAMPLE
O.g.f.: A(x) = 1 + 2*x + 10*x^2 + 106*x^3 + 1736*x^4 + 38414*x^5 + 1073178*x^6 +...
where
A(x) = 1 + 2*1^1*3^0*x*exp(-1*3*x) + 2*2^2*4^1*exp(-2*4*x)*x^2/2! + 2*3^3*5^2*exp(-3*5*x)*x^3/3! + 2*4^4*6^3*exp(-4*6*x)*x^4/4! + 2*5^5*7^4*exp(-5*7*x)*x^5/5! +...
simplifies to a power series in x with integer coefficients.
MATHEMATICA
Flatten[{1, Table[Sum[Binomial[n-1, j]*2^(n-j)*StirlingS2[n+j, n], {j, 0, n-1}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 22 2014 *)
PROG
(PARI) {a(n)=polcoeff(sum(m=0, n, 2*m^m*(m+2)^(m-1)*x^m*exp(-m*(m+2)*x+x*O(x^n))/m!), n)}
(PARI) {a(n)=(1/n!)*polcoeff(sum(k=0, n, 2*k^k*(k+2)^(k-1)*x^k/(1+k*(k+2)*x +x*O(x^n))^(k+1)), n)}
(PARI) {a(n)=1/n!*sum(k=0, n, 2*(-1)^(n-k)*binomial(n, k)*k^n*(k+2)^(n-1))}
(PARI) {a(n)=polcoeff(1+2*x*(1+2*x)^(n-1)/prod(k=0, n, 1-k*x +x*O(x^n)), n)}
(PARI) {a(n)=polcoeff(1+2*x*(1-2*x)^n/prod(k=0, n, 1-(k+2)*x +x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 14 2012
STATUS
approved