OFFSET
0,2
COMMENTS
Central coefficients of (1+2x-x^2)^n. Binomial transform of A098331.
Diagonal of rational function 1/(1 - (x^2 + 2*x*y - y^2)). - Gheorghe Coserea, Aug 04 2018
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
FORMULA
From Paul Barry, Sep 08 2004: (Start)
E.g.f. : exp(2*x)*BesselI(0, 2*I*x), I=sqrt(-1);
a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*binomial(n-k,k)*2^n*(-4)^(-k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*binomial(2(n-k),k)*(-2)^k. (End)
E.g.f. : exp(2*x)*BesselJ(0, 2*x). - Sergei N. Gladkovskii, Aug 22 2012
It appears that a(j+2) = (2*(2*j+1)*a(j+1))/(j+1)-(8*j*a(j))/(j+1), in case of re-indexing from 0 to 1. - Alexander R. Povolotsky, Aug 22 2012
D-finite with recurrence: a(n+2) = ((4*n+6)*a(n+1) - 8*(n+1)*a(n))/(n+2); a(0)=1,a(1)=2. - Sergei N. Gladkovskii, Aug 22 2012
a(n) = 2^n*_2F_1(1/2-n/2, -n/2, 1, -1), where _2F_1(a,b;c;x) is the hypergeometric function. - Alexander R. Povolotsky, Aug 22 2012
MATHEMATICA
a[n_] := 2^n*Hypergeometric2F1[1/2 - n/2, -n/2, 1, -1]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Sep 05 2012, after Alexander R. Povolotsky *)
CoefficientList[Series[1/Sqrt[1 - 4*x + 8*x^2], {x, 0, 50}], x] (* G. C. Greubel, Feb 19 2017 *)
PROG
(PARI) x='x+O('x^25); Vec(1/sqrt(1 - 4*x + 8*x^2)) \\ G. C. Greubel, Feb 19 2017
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul Barry, Sep 03 2004
STATUS
approved