OFFSET
0,2
FORMULA
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^2+k+1)^k * x^k / (1 + (k^2+k+1)*x)^(k+1).
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (k^2+k+1)^n.
EXAMPLE
O.g.f.: A(x) = 1 + 2*x + 16*x^2 + 208*x^3 + 3930*x^4 + 97956*x^5 +...
where
A(x) = exp(-x) + 3*x*exp(-3*x) + 7^2*exp(-7*x)*x^2/2! + 13^3*exp(-13*x)*x^3/3! + 21^4*exp(-21*x)*x^4/4! + 31^5*exp(-31*x)*x^5/5! +...
is a power series in x with integer coefficients.
PROG
(PARI) {a(n)=polcoeff(sum(k=0, n, (k^2+k+1)^k*exp(-(k^2+k+1)*x +x*O(x^n))*x^k/k!), n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n)=(1/n!)*polcoeff(sum(k=0, n, (k^2+k+1)^k*x^k/(1+(k^2+k+1)*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(k^2+k+1)^n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 24 2013
STATUS
approved