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

1, 1, 7, 55, 601, 7136, 116929, 1985607, 42814321, 954103114, 25933795801, 717297529686, 23297606120881, 770246625909788, 29137514248718373, 1127405063005559911, 48661170952876980481, 2139268956511467712586, 104127343895537535804841, 5158373570420037618149856

Offset

1,3

Links

Table of n, a(n) for n=1..20.

Formula

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.

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.

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).

Example

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!)*...]
where W(x) = 1 + x + 3*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! + 6^4*x^5/5! + ...
and W(x/exp(x)) = exp(x) and exp(x*W(x)) = W(x) = LambertW(-x)/(-x).

Mathematica

a[n_] :=  a[n] = n^(n-1) - (n-1)! * DivisorSum[n, #*(a[#]/#!)^(n/#) &, #<n &]; Array[a, 20] (* Amiram Eldar, Aug 18 2023 *)

Prog

(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)))))}
(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))}

Crossrefs

Cf. A159310.

Sequence in context: A112243 A083836 A326885 * A318580 A054910 A028562

Adjacent sequences: A159310 A159311 A159312 * A159314 A159315 A159316

Keyword

nonn

Author

Vladeta Jovovic and Paul D. Hanna, Apr 17 2009

Extensions

a(19)-a(20) from Amiram Eldar, Aug 18 2023

Status

approved