OFFSET
0,4
COMMENTS
Hankel transform is A055209.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..450
Shirali Kadyrov, Farukh Mashurov, Generalized continued fraction expansions for Pi and E, arXiv:1912.03214 [math.NT], 2019.
FORMULA
a(n+2) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = A001563(k+1) for k = 0, 1, ..., n.
E.g.f.: exp(x) / (1 + x).
a(n) = (-1)^n * A000166(n).
G.f.: 1/U(0) where U(k)= 1 - x + x*(k+1)/(1 + x*(k+1)/U(k+1)); (continued fraction). - Sergei N. Gladkovskii, Oct 13 2012
E.g.f.: exp(x) / (1 + x) = 1/(1 - x^2/2 + x^3/(U(0) + 2*x)) where U(k)= k^2 + k*(4-x) - 2*x + 3 + x*(k+1)*(k+3)^2/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 16 2012
G.f.: 1/Q(0) where Q(k) =1 + 4*k*x - x^2*(2*k + 1)^2/( 1 + (4*k+2)*x - x^2*(2*k + 2)^2/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 10 2013
G.f.: 1/Q(0), where Q(k)= 1 + (k+1)^2*(x) - x/(1 - x*(k+1)^2/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 18 2013
a(n) = (-1)^n*Gamma(n+1,-1)*exp(-1), where Gamma(a,x) is the incomplete gamma function. - Ilya Gutkovskiy, May 05 2016
0 = a(n)*(-a(n+1) +a(n+2) +a(n+3)) +a(n+1)*(+a(n+1) -2*a(n+2) -a(n+3)) +a(n+2)*(+a(n+2)) for all n>=0. - Michael Somos, Jun 26 2018
EXAMPLE
G.f. = 1 + x^2 - 2*x^3 + 9*x^4 - 44*x^5 + 265*x^6 - 1854*x^7 + 14833*x^8 + ...
MAPLE
a:= proc(n) option remember; `if`(n=0, 1, 1-n*a(n-1)) end:
seq(a(n), n=0..30); # Alois P. Heinz, Jul 19 2015
MATHEMATICA
a[n_] := x D[1/x Exp[x], {x, n}] x^n Exp[-x]
Table[a[n] /. x -> 1, {n, 0, 20}] (* Gerry Martens , May 05 2016 *)
a[0] = 1; a[n_] := a[n] = 1 - n a[n - 1]; Table[a@ n, {n, 0, 22}] (* Michael De Vlieger, May 05 2016 *)
PROG
(PARI) {a(n) = if( n<1, n==0, 1 - n * a(n-1))};
(Sage)
A182386 = lambda n: hypergeometric([-n, 1], [], 1)
print([simplify(A182386(n)) for n in range(23)]) # Peter Luschny, Oct 19 2014
(MAGMA) m:=35; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x)/(1+x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 11 2018
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Apr 27 2012
STATUS
proposed