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 A159310 G.f.: Product_{n>=1} (1 + a(n)*x^n/n!) = Sum_{n>=0} (n+1)^(n-1)*x^n/n! = LambertW(-x)/(-x). 1
 1, 3, 7, 97, 601, 7576, 116929, 2482537, 42814321, 1040362966, 25933795801, 760154969850, 23297606120881, 816970034324900, 29137514248718373, 1194044411689941241, 48661170952876980481, 2227962859999303395766 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS FORMULA G.f.: Sum_{n>=1} log(1 + a(n)*x^n/n!) = Sum_{n>=1} n^(n-1)*x^n/n! = -LambertW(-x). G.f.: Sum_{n>=1} log(1 + a(n)*exp(-n*x)*x^n/n!) = x. From Paul D. Hanna, Apr 15 2009: (Start) G.f.: Sum_{n>=1} n*a(n)*x^n/(n! + a(n)*x^n) = Sum_{n>=1} n^n*x^n/n!. G.f.: Sum_{n>=1} n*a(n)*x^n/(n!*exp(nx) + a(n)*x^n) = x/(1-x). Recurrence: a(n) = n^(n-1) + (n-1)!*((-1)^n + Sum_{d|n, 1 1 with a(1)=1. (End) EXAMPLE G.f.: W(x) = (1+x)*(1+3*x^2/2!)*(1+7*x^3/3!)*(1+97*x^4/4!)*(1+601*x^5/5!)* ... W(x) = 1 + x + 3*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! + 6^4*x^5/5! + ... where W(x/exp(x)) = exp(x) and exp(x*W(x)) = W(x) = LambertW(-x)/(-x). PROG (PARI) {a(n)=if(n<1, 0, polcoeff(sum(k=0, n, (k+1)^(k-1)*x^k/k!)/prod(k=1, n-1, 1+a(k)*x^k +x*O(x^n)), n))} (PARI) {a(n)=if(n<1, 0, if(n==1, 1, n^(n-1) + (n-1)!*((-1)^n + sumdiv(n, d, if(d1, d*(-a(d)/d!)^(n/d))))))} \\ Paul D. Hanna, Apr 15 2009 CROSSREFS Cf. A137852. Sequence in context: A088419 A062592 A074349 * A299377 A129660 A243734 Adjacent sequences:  A159307 A159308 A159309 * A159311 A159312 A159313 KEYWORD nonn AUTHOR Paul D. Hanna, Apr 15 2009 STATUS approved

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Last modified November 27 11:48 EST 2021. Contains 349385 sequences. (Running on oeis4.)