OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: 2*x + sqrt( 1 + 4*x^2 ) = 1 / (1 - 2*x / (1 + x / (1 - x / (1 + x / ... )))).
The g.f. A(x) satisfies: A(x) = sqrt(1 + 4*x * A(x)).
Conjecture : n*(n+1)*a(n) + (n+2)*(n-1)*a(n-1) +4*(n+1)*(n-3)*a(n-2) +4*(n+2)*(n-4)*a(n-3) = 0.- R. J. Mathar, Jul 24 2012
a(n) = 2*hypergeom([-n+1,2-n],[2],-1). - Peter Luschny, Sep 23 2014
0 = a(n)*(+16*a(n+2) + 10*a(n+4)) + a(n+2)*(-2*a(n+2) + a(n+4)) if n>=0. - Michael Somos, Jan 10 2017
a(n+4) = 2 * a(n+2) * (a(n+2) - 8*a(n)) / (a(n+2) + 10*a(n)) if n>=0 is even. - Michael Somos, Jan 10 2017
EXAMPLE
G.f. = 1 + 2*x + 2*x^2 - 2*x^4 + 4*x^6 - 10*x^8 + 28*x^10 - 84*x^12 + ...
MAPLE
s := proc(n) option remember; `if`(n<2, n+1, -4*(n-2)*s(n-2)/(n+1)) end: A127846 := n -> `if`(n<2, n+1, s(n-1)); seq(A127846(n), n=0..47); # Peter Luschny, Sep 23 2014
MATHEMATICA
CoefficientList[Series[Exp[ArcSinh[2x]], {x, 0, 50}], x] (* Harvey P. Dale, Aug 18 2012 *)
Table[2 HypergeometricPFQ[{-n+1, 2-n}, {2}, -1], {n, 0, 46}] (* Peter Luschny, Sep 23 2014 *)
PROG
(PARI) {a(n) = if( n<2, (n>=0) + (n>0), n = n-2; if( n%2, 0, (-1)^(n/2) * 4 * binomial( n, n/2) / (n + 2)))};
(PARI) {a(n) = if( n<0, 0, polcoeff( sqrt( 1 + 4*x^2 + x*O(x^n) ) + 2*x, n ) )};
(PARI) {a(n) = my(A); if( n<0, 0, A = 1 + O(x); for( k=1, n, A = sqrt( 1 + 4*x * A)); polcoeff( A, n))};
(Sage)
def A182122(n):
if n < 2: return n+1
if n % 2 == 1: return 0
return (-1)^(n/2-1)*binomial(n, n/2)/(n-1)
[A182122(n) for n in range(47)] # Peter Luschny, Sep 23 2014
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Exp(Argsinh(2*x)))); // G. C. Greubel, Aug 12 2018
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Apr 13 2012
STATUS
approved