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%I #26 Oct 09 2018 15:14:05
%S 1,1,3,22,293,6056,175687,6719476,325741705,19470659968,1403821003211,
%T 119836341280844,11923671362914093,1365089081187409072,
%U 177915120382062044815,26161941602115263558716,4306833594841510336897553,788302770933266249649820544,159446049770474152196515579027
%N E.g.f.: Sum_{n>=0} Series_Reversion( x/exp(n*x) )^n / n!.
%C LambertW identities utilized in the e.g.f.:
%C (1) Series_Reversion( x/exp(t*x) )^n = Sum_{k>=0} n*(n+k)^(k-1) * t^k * x^(n+k) / k!.
%C (2) Sum_{n>=0} Series_Reversion( x/exp(t*x) )^n/n! = Sum_{k>=0} (k*t+1)^(k-1)*x^k/k!.
%F a(n) = Sum_{k=0..n} binomial(n,k) * n^(k-1) * (n-k)^(k+1) for n>0 with a(0)=1.
%F 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)!.
%F E.g.f.: Sum_{n>=0} (-LambertW(-n*x))^n / (n^n * n!).
%F a(n) = [x^n] Sum_{k>=0} x^k/(1 - n*k*x)^k. - _Ilya Gutkovskiy_, Oct 09 2018
%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 293*x^4/4! + 6056*x^5/5! +...
%e where the series
%e A(x) = Sum_{n>=0} Series_Reversion( x/exp(n*x) )^n / n!
%e begins:
%e A(x) = 1 + (x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 625*x^5/5! +...)
%e + (x^2/2! + 12*x^3/3! + 192*x^4/4! + 4000*x^5/5! + 103680*x^6/6! +...)
%e + (x^3/3! + 36*x^4/4! + 1350*x^5/5! + 58320*x^6/6! +...)
%e + (x^4/4! + 80*x^5/5! + 5760*x^6/6! + 439040*x^7/7! +...)
%e + (x^5/5! + 150*x^6/6! + 18375*x^7/7! + 2240000*x^8/8! +...)
%e + (x^6/6! + 252*x^7/7! + 48384*x^8/8! + 8817984*x^9/9! +...)
%e + (x^7/7! + 392*x^8/8! + 111132*x^9/9! + 28812000*x^10/10! +...) +...
%e and equals
%e A(x) = Sum_{n>=0} Sum_{k>=0} C(n+k,k) * (n+k)^(k-1) * n^(k+1) * x^(n+k)/(n+k)!
%e = Sum_{n>=0} 1/n! * Sum_{k>=0} n*(n+k)^(k-1) * n^k * x^(n+k) / k!
%e = Sum_{n>=0} 1/n! * Series_Reversion( x/exp(n*x) )^n
%e = Sum_{n>=0} x^n/n! * Sum_{k=0..n} C(n,k) * n^(k-1) * (n-k)^(k+1).
%t 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 *)
%o (PARI) {a(n)=if(n==0,1,sum(k=0,n,binomial(n,k) * n^(k-1) * (n-k)^(k+1)))}
%o for(n=0,20,print1(a(n),", "))
%o (PARI) {a(n)=n!*polcoeff(1+sum(m=1,n,serreverse(x/exp(m*x +x*O(x^n)))^m/m!),n)}
%o for(n=0,20,print1(a(n),", "))
%o (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)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A299426.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 28 2014
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