A244468 E.g.f.: Sum_{n>=0} Series_Reversion( x/exp(n*x) )^n / n!.
1, 1, 3, 22, 293, 6056, 175687, 6719476, 325741705, 19470659968, 1403821003211, 119836341280844, 11923671362914093, 1365089081187409072, 177915120382062044815, 26161941602115263558716, 4306833594841510336897553, 788302770933266249649820544, 159446049770474152196515579027
Offset: 0
Keywords
Examples
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).
Crossrefs
Cf. A299426.
Programs
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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 *) -
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), ", "))
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
Comments