A222523 O.g.f.: Sum_{n>=0} (n^2+n+1)^n * exp(-(n^2+n+1)*x) * x^n / n!.

1, 2, 16, 208, 3930, 97956, 3038968, 112911296, 4889301222, 241822567180, 13450863716048, 831128810329632, 56483233790927556, 4187162929534240488, 336244786874092579920, 29077531985735270053632, 2694076376135933879002566, 266245292488900189811625612, 27956094249950913890814701248

Offset

0,2

Links

Table of n, a(n) for n=0..18.

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

Cf. A217900, A217901.

Sequence in context: A161568 A138429 A087923 * A004121 A188600 A206930

Adjacent sequences: A222520 A222521 A222522 * A222524 A222525 A222526

Keyword

nonn

Author

Paul D. Hanna, Feb 24 2013

Status

approved