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 A244468 E.g.f.: Sum_{n>=0} Series_Reversion( x/exp(n*x) )^n / n!. 1
 1, 1, 3, 22, 293, 6056, 175687, 6719476, 325741705, 19470659968, 1403821003211, 119836341280844, 11923671362914093, 1365089081187409072, 177915120382062044815, 26161941602115263558716, 4306833594841510336897553, 788302770933266249649820544, 159446049770474152196515579027 (list; graph; refs; listen; history; text; internal format)
 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!. LINKS 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 Cf. A299426. Sequence in context: A122778 A108991 A247659 * A325295 A298693 A326430 Adjacent sequences:  A244465 A244466 A244467 * A244469 A244470 A244471 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 28 2014 STATUS approved

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Last modified December 6 15:57 EST 2021. Contains 349565 sequences. (Running on oeis4.)