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A098331
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Expansion of 1/sqrt(1-2x+5x^2).
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12
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1, 1, -1, -5, -5, 11, 41, 29, -125, -365, -131, 1409, 3301, -155, -15625, -29485, 16115, 170035, 254525, -309775, -1813055, -2064655, 4617755, 18909175, 14903725, -61552739, -192390589, -81290561, 767919595, 1901796395, 28588201
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OFFSET
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0,4
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COMMENTS
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Central coefficients of (1+x-x^2)^n. Binomial transform of 1/sqrt(1+4x^2), or (1,0,-2,0,6,0,-20,...). Binomial transform is A098335. (-1)^nA098331(n) is the inverse binomial transform of (1,0,-2,0,6,0,-20,...).
Hankel transform is 2^n*(-1)^C(n+1,2). Hankel transform of 0,1,1,-1,-5,-5,... is F(n)*(-1)^C(n+2,2)*(2^n+0^n)/2. - Paul Barry, Jan 13 2009
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LINKS
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Robert Israel, Table of n, a(n) for n = 0..2865
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
Eric Weisstein's World of Mathematics, Trinomial Coefficient
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FORMULA
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E.g.f.: exp(x)BesselI(0, 2*I*x), I=sqrt(-1);
a(n) = sum{k=0..floor(n/2), binomial(n, 2k)*binomial(2k, k)(-1)^k};
a(n) = sum{k=0..floor(n/2), binomial(n, k)*binomial(n-k, k)(-1)^k);
a(n) = sum{k=0..n, binomial(n, k)binomial(k, k/2)cos(pi*k/2)}.
D-finite: a(0)=a(1)=1, a(n) = ((2n-1)a(n-1)-5(n-1)a(n-2))/n. - T. D. Noe, Oct 19 2005
a(n) = hypergeom([-n/2, 1/2-n/2], [1], -4). - Peter Luschny, Sep 18 2014
a(n) ~ 5^(n/2 + 1/4) * cos((Pi*n - arctan(1/2) - n*arctan(4/3))/2) / sqrt(Pi*n). - Vaclav Kotesovec, Oct 31 2017
a(n) = (sqrt(5))^n*P(n,1/sqrt(5)), where P(n,x) is the Legendre polynomial of degree n. Note the general result (sqrt(4*m+1))^n*P(n, 1/sqrt(4*m+1)) = Sum_{k = 0..floor(n/2)} C(n,2*k)*C(2*k,k)(-m)^k due to Catalan. - Peter Bala, Mar 18 2018
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MAPLE
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A098331 := n -> hypergeom([-n/2, 1/2-n/2], [1], -4);
seq(round(evalf(A098331(n), 99)), n=0..30); # Peter Luschny, Sep 18 2014
f:= gfun:-rectoproc({(5*n+5)*a(n)+(-3-2*n)*a(n+1)+(n+2)*a(n+2), a(0) = 1, a(1) = 1}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Jan 30 2018
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MATHEMATICA
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a=b=1; Join[{a, b}, Table[c=((2n-1)b-5(n-1)a)/n; a=b; b=c; c, {n, 2, 30}]] (Noe)
CoefficientList[Series[1/Sqrt[1-2x+5x^2], {x, 0, 40}], x] (* Harvey P. Dale, Aug 17 2015 *)
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PROG
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(PARI) x='x+O('x^99); Vec(1/(1-2*x+5*x^2)^(1/2)) \\ Altug Alkan, Mar 18 2018
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CROSSREFS
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Cf. A098332, A098333, A098334.
Sequence in context: A299591 A173316 A242325 * A222575 A222681 A061391
Adjacent sequences: A098328 A098329 A098330 * A098332 A098333 A098334
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KEYWORD
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easy,sign,changed
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AUTHOR
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Paul Barry, Sep 03 2004
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EXTENSIONS
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Corrected by T. D. Noe, Oct 19 2005
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STATUS
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approved
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