OFFSET
0,3
LINKS
Robert Israel, Table of n, a(n) for n = 0..448
FORMULA
(1-x)^(-1-x) = Sum_{n>=0} (Product_{k=1..n} (k+x)) * x^n/n!. [Paul D. Hanna, Nov 01 2010]
E.g.f.: (1-x)^(-1-x) = 1+(x*(1+x))/(Q(0)-x*(1+x)); Q(k)=(1+x)*k+1+x+(x^2)-x*(k+1)*(k+2+x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 27 2011
E.g.f.: 1 + x*(Q(0) - 1)/(x-1) where Q(k) = 1 - (1+x/(k+1))/(1 - x/(x - 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 05 2013
a(n) ~ n! * (n - log(n) + 1 - gamma), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Sep 29 2013
E.g.f.: exp( x + Sum_{n>=2} (2*n-1) * x^n / (n*(n-1)) ). - Paul D. Hanna, Sep 27 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 15*x^3/3! + 80*x^4/4! + 490*x^5/5! +...
Explicit expressions for the e.g.f.:
(1-x)^(-1-x) = 1 + (1+x)*x + (1+x)(2+x)*x^2/2! + (1+x)(2+x)(3+x)*x^3/3! +... - Paul D. Hanna, Nov 01 2010
(1-x)^(-1-x) = exp(x + 3*x^2/2 + 5*x^3/6 + 7*x^4/12 + 9*x^5/20 + 11*x^6/30 +...). - Paul D. Hanna, Sep 27 2014
MAPLE
S:= series((1-x)^(-1-x), x, 51):
seq(coeff(S, x, j)*j!, j=0..50); # Robert Israel, Apr 20 2017
MATHEMATICA
CoefficientList[ Series[(1 - x)^(-1 - x), {x, 0, 19}], x]*Table[(n - 1)!, {n, 1, 20}]
PROG
(PARI) {a(n)=n!*polcoeff(sum(m=0, n, prod(k=1, m, k+x)*x^m/m!)+x*O(x^n), n)} \\ Paul D. Hanna, Nov 01 2010
(PARI) {a(n)=n!*polcoeff((1-x+x*O(x^n))^(-1-x), n)} \\ Paul D. Hanna, Nov 01 2010
(Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1-x)^(-1-x) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 30 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Aug 26 2002
STATUS
approved