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A124824 LambertW analog of the Bell numbers: a(n) = (1/e)*Sum_{k>=0} k*(n+k)^(n-1)/k! for n > 0 with a(0)=1. 1
1, 1, 4, 26, 235, 2727, 38699, 649931, 12616132, 278054700, 6861571205, 187474460527, 5619443518165, 183375548287557, 6472290237774352, 245705256222934490, 9983967457086797107, 432392173830077506403 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

FORMULA

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 D. Hanna, Nov 10 2006

a(n) ~ exp(exp(1))*n^(n-1). - Vaclav Kotesovec, Jan 04 2013

a(n+1) = Sum_{k = 0..n} binomial(n,k)*(n - k + 2)^(n-k)*a(k) with a(0) = 1. - Peter Bala, Nov 21 2016

EXAMPLE

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.

MAPLE

A124824 := proc (n) option remember;

if n = 0 then

  1;

else

  add(binomial(n-1, k)*(n-k+1)^(n-1-k)*A124824(k), k = 0..n-1);

end if;

end proc:

seq(A124824(n), n = 0..20);

# Peter Bala, Nov 22 2016

MATHEMATICA

Flatten[{1, Table[Sum[Binomial[n-1, k-1]*n^(n-k)*BellB[k], {k, 1, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jan 04 2013 *)

PROG

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)))}

CROSSREFS

Cf. A000272, A000110.

Sequence in context: A136227 A000310 A054360 * A000311 A244451 A001863

Adjacent sequences:  A124821 A124822 A124823 * A124825 A124826 A124827

KEYWORD

nonn,easy

AUTHOR

Paul D. Hanna, Nov 09 2006

STATUS

approved

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Last modified February 18 15:30 EST 2020. Contains 332019 sequences. (Running on oeis4.)