A217905 O.g.f.: Sum_{n>=0} -n^n*(n-1)^(n-1) * exp(-n*(n-1)*x) * x^n / n!.

1, -1, -2, -14, -184, -3532, -89256, -2800016, -104967808, -4578528464, -227816059360, -12735645181536, -790296855912576, -53905019035510528, -4008716449677965312, -322807879692969879552, -27983800239966141382656, -2598368754552749176202496, -257284990746988090769530368

Offset

0,3

Comments

Compare the g.f. to the LambertW identity:

1 = Sum_{n>=0} -(n-1)^(n-1) * exp(-(n-1)*x) * x^n/n!.

Links

G. C. Greubel, Table of n, a(n) for n = 0..345

Formula

a(n) = -A191236(n-1) for n>=1. [corrected by Vaclav Kotesovec, Aug 22 2018]

a(n) = 1/n! * Sum_{k=0..n} -(-1)^(n-k)*binomial(n,k) * k^n * (k-1)^(n-1) for n>=0.

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

a(n) = [x^n] 1 - x*(1-x)^(n-1) / Product_{k=1..n} (1-k*x).

a(n) = [x^n] 1 - x*(1+x)^(n-1) / Product_{k=1..n} (1-(k-1)*x).

a(n) ~ -2^(n-1) * exp(n*(r-1)-r) * n^(n - 3/2) / (sqrt(Pi*(r-1)*(2-r)) * r^(n-1)), where r = 2 + LambertW(-2*exp(-2)) = A256500 = 1.5936242600400400923230418... - Vaclav Kotesovec, Aug 22 2018

Example

O.g.f.: A(x) = 1 - x - 2*x^2 - 14*x^3 - 184*x^4 - 3532*x^5 - 89256*x^6 +...
where
A(x) = 1 - 1^1*0^0*x*exp(-1*0*x) - 2^2*1^1*exp(-2*1*x)*x^2/2! - 3^3*2^2*exp(-3*2*x)*x^3/3! - 4^4*3^3*exp(-4*3*x)*x^4/4! - 5^5*4^4*exp(-5*4*x)*x^5/5! +...
simplifies to a power series in x with integer coefficients.

Mathematica

Join[{1, -1}, Table[(1/n!)*Sum[(-1)^(n - k + 1)*Binomial[n, k]*k^n*(k - 1)^(n - 1), {k, 0, n}], {n, 2, 50}]] (* G. C. Greubel, Nov 16 2017 *)

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, -m^m*(m-1)^(m-1)*x^m*exp(-m*(m-1)*x+x*O(x^n))/m!), n)}
(PARI) {a(n)=(1/n!)*polcoeff(sum(k=0, n, -k^k*(k-1)^(k-1)*x^k/(1+k*(k-1)*x +x*O(x^n))^(k+1)), n)}
(PARI) {a(n)=1/n!*sum(k=0, n, -(-1)^(n-k)*binomial(n, k)*k^n*(k-1)^(n-1))}
(PARI) {a(n)=polcoeff(1-x*(1-x)^(n-1)/prod(k=0, n, 1-k*x +x*O(x^n)), n)}
(PARI) {a(n)=polcoeff(1-x*(1+x)^n/prod(k=0, n, 1-(k-1)*x +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A191236, A217906, A217899, A217900, A217901, A217902, A217903, A217904.

Sequence in context: A000807 A191565 A191236 * A370940 A237852 A219874

Adjacent sequences: A217902 A217903 A217904 * A217906 A217907 A217908

Keyword

sign

Author

Paul D. Hanna, Oct 14 2012

Status

approved