login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 3
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
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
Sequence in context: A346978 A000310 A054360 * A000311 A244451 A001863
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Nov 09 2006
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)