A098331 Expansion of 1/sqrt(1 - 2*x + 5*x^2).

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

Offset

0,4

Comments

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

Links

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

Formula

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 with recurrence: 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

G.f.: 1/(1 - x + 2*x^2/(1 - x + x^2/(1 - x + x^2/(1 - x + x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Nov 19 2021

From Peter Bala, Feb 08 2022: (Start)

G.f.: A(x) = Sum_{n >= 0} (-1)^n*binomial(2*n,n)*x^(2*n)/(1 - x)^(2*n+1).

a(n)^2 = Sum_{k = 0..n} (-1)^k*5^(n-k)*binomial(2*k,k)*binomial(n,k)* binomial(n+k,k).

Sum_{n >= 0} (-1)^n*binomial(2*n,n)^2 * x^n/(1 - 5*x)^(2*n+1) = 1 + x + x^2 + 25*x^3 + 25*x^4 + 121*x^5 + ... is the g.f. of a(n)^2.

The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all prime p and positive integers n and k. (End)

Maple

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

Mathematica

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 *)

Prog

(PARI) my(x='x+O('x^99)); Vec(1/(1-2*x+5*x^2)^(1/2)) \\ Altug Alkan, Mar 18 2018

Crossrefs

Cf. A098332, A098333, A098334.

Sequence in context: A350920 A173316 A242325 * A222575 A222681 A061391

Adjacent sequences: A098328 A098329 A098330 * A098332 A098333 A098334

Keyword

easy,sign

Author

Paul Barry, Sep 03 2004

Extensions

Corrected by T. D. Noe, Oct 19 2005

Status

approved