OFFSET
0,3
COMMENTS
LambertW identities utilized in the e.g.f.:
(1) Series_Reversion( x/exp(t*x) )^n = Sum_{k>=0} n*(n+k)^(k-1) * t^k * x^(n+k) / k!.
(2) Sum_{n>=0} Series_Reversion( x/exp(t*x) )^n/n! = Sum_{k>=0} (k*t+1)^(k-1)*x^k/k!.
FORMULA
a(n) = Sum_{k=0..n} binomial(n,k) * n^(k-1) * (n-k)^(k+1) for n>0 with a(0)=1.
E.g.f.: Sum_{n>=0} Sum_{k>=0} C(n+k,k) * (n+k)^(k-1) * n^(k+1) * x^(n+k)/(n+k)!.
E.g.f.: Sum_{n>=0} (-LambertW(-n*x))^n / (n^n * n!).
a(n) = [x^n] Sum_{k>=0} x^k/(1 - n*k*x)^k. - Ilya Gutkovskiy, Oct 09 2018
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 293*x^4/4! + 6056*x^5/5! +...
where the series
A(x) = Sum_{n>=0} Series_Reversion( x/exp(n*x) )^n / n!
begins:
A(x) = 1 + (x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 625*x^5/5! +...)
+ (x^2/2! + 12*x^3/3! + 192*x^4/4! + 4000*x^5/5! + 103680*x^6/6! +...)
+ (x^3/3! + 36*x^4/4! + 1350*x^5/5! + 58320*x^6/6! +...)
+ (x^4/4! + 80*x^5/5! + 5760*x^6/6! + 439040*x^7/7! +...)
+ (x^5/5! + 150*x^6/6! + 18375*x^7/7! + 2240000*x^8/8! +...)
+ (x^6/6! + 252*x^7/7! + 48384*x^8/8! + 8817984*x^9/9! +...)
+ (x^7/7! + 392*x^8/8! + 111132*x^9/9! + 28812000*x^10/10! +...) +...
and equals
A(x) = Sum_{n>=0} Sum_{k>=0} C(n+k,k) * (n+k)^(k-1) * n^(k+1) * x^(n+k)/(n+k)!
= Sum_{n>=0} 1/n! * Sum_{k>=0} n*(n+k)^(k-1) * n^k * x^(n+k) / k!
= Sum_{n>=0} 1/n! * Series_Reversion( x/exp(n*x) )^n
= Sum_{n>=0} x^n/n! * Sum_{k=0..n} C(n,k) * n^(k-1) * (n-k)^(k+1).
MATHEMATICA
Flatten[{1, Table[Sum[Binomial[n, k] * n^(k-1) * (n-k)^(k+1), {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jul 13 2014 *)
PROG
(PARI) {a(n)=if(n==0, 1, sum(k=0, n, binomial(n, k) * n^(k-1) * (n-k)^(k+1)))}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=n!*polcoeff(1+sum(m=1, n, serreverse(x/exp(m*x +x*O(x^n)))^m/m!), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(LambertW=serreverse(x*exp(x+x*O(x^n))), A=1); A=sum(m=0, n, 1/m!/m^m*subst(-LambertW, x, -m*x)^m); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 28 2014
STATUS
approved