|
Search: id:A159310
|
|
|
| A159310 |
|
G.f.: Product_{n>=1} (1 + a(n)*x^n/n!) = Sum_{n>=0} (n+1)^(n-1)*x^n/n! = LambertW(-x)/(-x). |
|
+0
2
|
|
| 1, 3, 7, 97, 601, 7576, 116929, 2482537, 42814321, 1040362966, 25933795801, 760154969850, 23297606120881, 816970034324900, 29137514248718373, 1194044411689941241, 48661170952876980481, 2227962859999303395766
(list; graph; listen)
|
|
|
OFFSET
|
1,2
|
|
|
FORMULA
|
G.f.: Sum_{n>=1} log(1 + a(n)*x^n/n!) = Sum_{n>=1} n^(n-1)*x^n/n! = -LambertW(-x).
G.f.: Sum_{n>=1} log(1 + a(n)*exp(-n*x)*x^n/n!) = x.
Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Apr 15 2009: (Start)
G.f.: Sum_{n>=1} n*a(n)*x^n/[n! + a(n)*x^n] = Sum_{n>=1} n^n*x^n/n!.
G.f.: Sum_{n>=1} n*a(n)*x^n/[n!*exp(nx) + a(n)*x^n] = x/(1-x).
Recurrence:
a(n) = n^(n-1) + (n-1)!*[(-1)^n + Sum_{d divides n, 1<d<n} d*( -a(d)/d! )^(n/d) ] for n>1 with a(1)=1.
(End)
|
|
EXAMPLE
|
G.f.: W(x) = (1+x)*(1+3*x^2/2!)*(1+7*x^3/3!)*(1+97*x^4/4!)*(1+601*x^5/5!)*...
W(x) = 1 + x + 3*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! + 6^4*x^5/5! +...
where W(x/exp(x)) = exp(x) and exp(x*W(x)) = W(x) = LambertW(-x)/(-x).
|
|
PROGRAM
|
(PARI) {a(n)=if(n<1, 0, polcoeff(sum(k=0, n, (k+1)^(k-1)*x^k/k!)/prod(k=1, n-1, 1+a(k)*x^k +x*O(x^n)), n))}
(PARI) {a(n)=if(n<1, 0, if(n==1, 1, n^(n-1) + (n-1)!*((-1)^n + sumdiv(n, d, if(d<n&d>1, d*(-a(d)/d!)^(n/d))))))} [From Paul D. Hanna (pauldhanna(AT)juno.com), Apr 15 2009]
|
|
CROSSREFS
|
Cf. A137852.
Sequence in context: A088419 A062592 A074349 this_sequence A129660 A158467 A028414
Adjacent sequences: A159307 A159308 A159309 this_sequence A159311 A159312 A159313
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Paul D. Hanna (pauldhanna(AT)juno.com), Apr 15 2009
|
|
|
Search completed in 0.002 seconds
|
