OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: 1 / ( 1 + x / (1 + 4*x / (1 - x^2 / (1 + 4*x / (1 - x^2 / (1 + 4*x / ...)))))). - Michael Somos, Jan 03 2013
a(n) = (-1)^n*Sum_{k=0..n} A253283(n,k). - Peter Luschny, Mar 23 2015
From Peter Bala, Jun 17 2015: (Start)
a(n) = Sum_{k = 0..n} (-2)^k*binomial(n,k)*binomial(n+k-1,k) = (-1)^n*Sum_{k = 0..n-1} binomial(n,k+1)*binomial(n+k,k) = -Sum_{k = 0..n-1} (-2)^k*binomial(n-1,k)*binomial(n+k,k).
a(n) = -R(n-1,-2) for n >= 1, where R(n,x) denotes the n-th row polynomial of A178301.
a(n) = [x^n] ((x - 1)/(1 - 2*x))^n. Cf. A001003(n) = (-1)^(n+1)/(n+1)*[x^n] ((x - 1)/(1 - 2*x))^(n+1).
O.g.f.: 1/2 + 1/2*(1 + x)/sqrt(1 + 6*x + x^2).
exp( Sum_{n >= 1} a(n)*(-x)^n/n ) = 1 + x + 3*x^2 + 11*x^3 + 45*x^4 + ... is the o.g.f. for A001003.
Recurrence: n*(3 - 2*n )*a(n) = 2*(6*n^2 - 12*n + 5)*a(n-1) + (2*n - 1)*(n - 2)*a(n-2) with a(0) = 1, a(1) = -1. (End)
EXAMPLE
G.f. = 1 - x + 5*x^2 - 25*x^3 + 129*x^4 - 681*x^5 + 3653*x^6 - 19825*x^7 + ...
MAPLE
seq(simplify(hypergeom([-n, n], [1], 2)), n=0..21); # Peter Luschny, Mar 23 2015
MATHEMATICA
a[n_] := Hypergeometric2F1[ -n, n, 1, 2]; Array[a, 20, 0]
PROG
(PARI) {a(n) = sum( k=0, abs(n), 2^k * prod( i=0, k-1, i^2 - n^2 ) / k!^2)}
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/2 + 1/2*(1 + x)/Sqrt(1 + 6*x + x^2))); // G. C. Greubel, Aug 14 2018
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Michael Somos, Feb 06 2011
STATUS
approved