A222524 O.g.f.: Sum_{n>=0} n^n*(2*n+1)^n * exp(-n*(2*n+1)*x) * x^n / n!.

1, 3, 41, 1057, 40057, 2006631, 125093285, 9333786225, 811181004929, 80480710535035, 8975976702322401, 1111688368710017121, 151388120776146737641, 22482576760232188394991, 3616177985990080869347277, 626250139757797928093888481, 116181112230230754285955844865

Offset

0,2

Links

Table of n, a(n) for n=0..16.

Formula

a(n) = 1/n! * [x^n] Sum_{k>=0} k^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^n*(2*k+1)^n.

Example

 O.g.f.: A(x) = 1 + 3*x + 41*x^2 + 1057*x^3 + 40057*x^4 + 2006631*x^5 +...
where
A(x) = 1 + 3*x*exp(-3*x) + 10^2*exp(-10*x)*x^2/2! + 21^3*exp(-21*x)*x^3/3! + 36^4*exp(-36*x)*x^4/4! + 55^5*exp(-55*x)*x^5/5! +...
is a power series in x with integer coefficients.

Prog

 (PARI) {a(n)=polcoeff(sum(k=0, n, k^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^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^n*(2*k+1)^n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A217900, A217901.

Sequence in context: A300281 A012175 A007313 * A241704 A181675 A012053

Adjacent sequences: A222521 A222522 A222523 * A222525 A222526 A222527

Keyword

nonn

Author

Paul D. Hanna, Feb 24 2013

Status

approved