cp's OEIS Frontend

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

%I A217905 #24 Aug 22 2018 03:08:49
%S A217905 1,-1,-2,-14,-184,-3532,-89256,-2800016,-104967808,-4578528464,
%T A217905 -227816059360,-12735645181536,-790296855912576,-53905019035510528,
%U A217905 -4008716449677965312,-322807879692969879552,-27983800239966141382656,-2598368754552749176202496,-257284990746988090769530368
%N A217905 O.g.f.: Sum_{n>=0} -n^n*(n-1)^(n-1) * exp(-n*(n-1)*x) * x^n / n!.
%C A217905 Compare the g.f. to the LambertW identity:
%C A217905 1 = Sum_{n>=0} -(n-1)^(n-1) * exp(-(n-1)*x) * x^n/n!.
%H A217905 G. C. Greubel, <a href="/A217905/b217905.txt">Table of n, a(n) for n = 0..345</a>
%F A217905 a(n) = -A191236(n-1) for n>=1. [corrected by _Vaclav Kotesovec_, Aug 22 2018]
%F A217905 a(n) = 1/n! * Sum_{k=0..n} -(-1)^(n-k)*binomial(n,k) * k^n * (k-1)^(n-1) for n>=0.
%F A217905 a(n) = 1/n! * [x^n] Sum_{k>=0} -k^k*(k-1)^(k-1)*x^k / (1 + k*(k-1)*x)^(k+1).
%F A217905 a(n) = [x^n] 1 - x*(1-x)^(n-1) / Product_{k=1..n} (1-k*x).
%F A217905 a(n) = [x^n] 1 - x*(1+x)^(n-1) / Product_{k=1..n} (1-(k-1)*x).
%F A217905 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
%e A217905 O.g.f.: A(x) = 1 - x - 2*x^2 - 14*x^3 - 184*x^4 - 3532*x^5 - 89256*x^6 +...
%e A217905 where
%e A217905 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! +...
%e A217905 simplifies to a power series in x with integer coefficients.
%t A217905 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 *)
%o A217905 (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)}
%o A217905 (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)}
%o A217905 (PARI) {a(n)=1/n!*sum(k=0,n, -(-1)^(n-k)*binomial(n,k)*k^n*(k-1)^(n-1))}
%o A217905 (PARI) {a(n)=polcoeff(1-x*(1-x)^(n-1)/prod(k=0, n, 1-k*x +x*O(x^n)), n)}
%o A217905 (PARI) {a(n)=polcoeff(1-x*(1+x)^n/prod(k=0, n, 1-(k-1)*x +x*O(x^n)), n)}
%o A217905 for(n=0,30,print1(a(n),", "))
%Y A217905 Cf. A191236, A217906, A217899, A217900, A217901, A217902, A217903, A217904.
%K A217905 sign
%O A217905 0,3
%A A217905 _Paul D. Hanna_, Oct 14 2012