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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.
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%I #20 May 11 2019 11:43:01

%S 1,1,4,26,235,2727,38699,649931,12616132,278054700,6861571205,

%T 187474460527,5619443518165,183375548287557,6472290237774352,

%U 245705256222934490,9983967457086797107,432392173830077506403

%N 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.

%H Vincenzo Librandi, <a href="/A124824/b124824.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f.: A(x) = exp(L(x) - 1), where L(x) = -LambertW(-x)/x. - _Vladeta Jovovic_, Nov 10 2006

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

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

%F 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

%F a(n) ~ exp(exp(1))*n^(n-1). - _Vaclav Kotesovec_, Jan 04 2013

%F 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

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

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

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

%e 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:

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

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

%p A124824 := proc (n) option remember;

%p if n = 0 then

%p 1;

%p else

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

%p end if;

%p end proc:

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

%p # _Peter Bala_, Nov 22 2016

%t 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 *)

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

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

%o (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)))}

%Y Cf. A000272, A000110.

%K nonn,easy

%O 0,3

%A _Paul D. Hanna_, Nov 09 2006