cp's OEIS Frontend

A159313 G.f.: 1/Product_{n>=1} (1 - a(n)*x^n/n!) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.

%I A159313 #7 Aug 20 2023 10:50:26
%S A159313 1,1,7,55,601,7136,116929,1985607,42814321,954103114,25933795801,
%T A159313 717297529686,23297606120881,770246625909788,29137514248718373,
%U A159313 1127405063005559911,48661170952876980481,2139268956511467712586,104127343895537535804841,5158373570420037618149856
%N A159313 G.f.: 1/Product_{n>=1} (1 - a(n)*x^n/n!) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.
%F A159313 a(n) = n^(n-1) - (n-1)!*[ Sum_{d divides n, d<n} d*( a(d)/d! )^(n/d) ] for n>1 with a(1)=1.
%F A159313 G.f.: Sum_{n>=1} -log(1 - a(n)*x^n/n!) = Sum_{n>=1} n^(n-1)*x^n/n! = -LambertW(-x).
%F A159313 G.f.: Sum_{n>=1} -log(1 - a(n)*exp(-n*x)*x^n/n!) = x.
%F A159313 G.f.: Sum_{n>=1} n*a(n)*x^n/[n! - a(n)*x^n] = Sum_{n>=1} n^n*x^n/n!.
%F A159313 G.f.: Sum_{n>=1} n*a(n)*x^n/[n!*exp(nx) - a(n)*x^n] = x/(1-x).
%e A159313 G.f.: W(x) = 1/[(1-x)*(1-x^2/2!)*(1-7*x^3/3!)*(1-55*x^4/4!)*(1-601*x^5/5!)*...]
%e A159313 where W(x) = 1 + x + 3*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! + 6^4*x^5/5! + ...
%e A159313 and W(x/exp(x)) = exp(x) and exp(x*W(x)) = W(x) = LambertW(-x)/(-x).
%t A159313 a[n_] :=  a[n] = n^(n-1) - (n-1)! * DivisorSum[n, #*(a[#]/#!)^(n/#) &, #<n &]; Array[a, 20] (* _Amiram Eldar_, Aug 18 2023 *)
%o A159313 (PARI) {a(n)=if(n<1, 0, if(n==1, 1,n^(n-1) - (n-1)!*sumdiv(n, d, if(d<n, d*(a(d)/d!)^(n/d)))))}
%o A159313 (PARI) {a(n)=if(n<1, 0, n!*polcoeff(sum(k=0,n+1,(k+1)^(k-1)*x^k/k!)*prod(k=1, n-1, 1-a(k)*x^k/k! +x*O(x^n)), n))}
%Y A159313 Cf. A159310.
%K A159313 nonn
%O A159313 1,3
%A A159313 _Vladeta Jovovic_ and _Paul D. Hanna_, Apr 17 2009
%E A159313 a(19)-a(20) from _Amiram Eldar_, Aug 18 2023