0,3

E.g.f.: A(x) = exp(L(x) - 1), where L(x) = -LambertW(-x)/x. - Vladeta Jovovic, Nov 10 2006

E.g.f.: A(x) = exp( Sum_{n>=1} (n+1)^(n-1)*x^n/n! ).

a(n) = Sum_{k=0..n} C(n-1,k-1)*n^(n-k)*Bell(k).

More generally: e.g.f. B(x,m) = exp(L(x)^m - 1) generates the sequence: a(n) = Sum_{k=0..n}m^k* C(n-1,k-1)*n^(n-k)*Bell(k) and also a(n) = (1/e)*Sum_{k>=0} m*k*(n+m*k)^(n-1)/k! for n>0 with a(0)=1. - Vladeta Jovovic and Paul Hanna, Nov 10 2006

A(x) = 1 + x + 4*x^2/2! + 26*x^3/3! + 235*x^4/4! + 2727*x^5/5! +...

E.g.f.: log(A(x)) = L(x) - 1, where L(x) = -LambertW(-x)/x, or,

L(x) = 1 + x + 3*x^2/2! + 16*x^3/3! +...+ (n+1)^(n-1)*x^n/n! +...

Since L(x)^k = Sum_{n>=0} k*(n+k)^(n-1)*x^n/n!, for all k, then the series representation of the g.f. is derived from:

A(x) = (1/e)*Sum_{k>=0} Sum_{n>=0} k*(n+k)^(n-1)/k!*x^n/n!

so that a(n) = (1/e)*Sum_{k>=0} k*(n+k)^(n-1)/k! with a(0)=1.

a(n)=n!*polcoeff(exp(sum(m=0, n, (m+1)^(m-1)*x^m/m!)-1), n)

(PARI) {a(n)=if(n==0, 1, round(exp(-1)*sum(k=0, 3*n, k*(k+n)^(n-1)/k!)))}

(PARI) {a(n)=if(n==0, 1, sum(k=0, n, binomial(n-1, k-1)*n^(n-k)*k!* polcoeff(exp(exp(x+x*O(x^k))-1), k)))}

Cf. A000272, A000110.

Adjacent sequences: A124821 A124822 A124823 this_sequence A124825 A124826 A124827

Sequence in context: A136227 A000310 A054360 this_sequence A000311 A001863 A115416

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Nov 09 2006