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, 4, 26, 235, 2727, 38699, 649931, 12616132, 278054700, 6861571205, 187474460527, 5619443518165, 183375548287557, 6472290237774352, 245705256222934490, 9983967457086797107, 432392173830077506403
Offset: 0
Examples
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.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
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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
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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 *) -
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)))}
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