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A012000 Expansion of 1/sqrt(1 - 4*x + 16*x^2). 6
1, 2, -2, -28, -74, 92, 1324, 3656, -4826, -70228, -197372, 267896, 3921724, 11126936, -15347432, -225505648, -643622906, 897078476, 13214495764, 37869162392, -53170602284, -784672445368, -2255295815192, 3183829452272, 47051201187676, 135537088268792, -192142210448216 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

T. D. Noe, Table of n, a(n) for n = 0..200

Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.

Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

FORMULA

G.f.: 1/sqrt(1 - 4*x + 16*x^2). - Vladeta Jovovic, May 13 2003

Scaled Legendre polynomials evaluated at 1/2: 2^(2n)P(n, 1/2). - Michael Somos, Dec 03 2001

a(n) = (-1)^n*Sum_{k=0..n} C(n,k)^2*(-3)^k. - Benoit Cloitre, Oct 25 2003

a(n) = Sum_{k=0..floor(n/2)} C(n,k)*C(2(n-k),n)(-4)^k. - Paul Barry, Sep 08 2004

D-finite with recurrence: n*a(n) + 2*(1-2*n)*a(n-1) + 16*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011

G.f.: G(0), where G(k) = 1 + x*(1-x)*(4*k+1)/( 4*k+2 - x*(1-x)*(4*k+2)*(4*k+3)/(x*(1-x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 06 2013

a(n) ~ 2^(2*n+1)*sin(Pi/12*(4*n+5))/(3^(1/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Jul 30 2013

a(n) = [x^n] ((1 - x)*(1 + 3*x))^n. - Peter Bala, Aug 16 2015

a(n) = 4^n*hypergeom([-n,1+n], [1], 1/4). - Peter Luschny, May 09 2016

From Peter Bala, Nov 28 2021: (Start)

a(n) = 4^(2*n+1)*Sum_{k >= n} (-1)^(n+k)*C(k,n)^2*(1/3)^(k+1).

a(n) = (-1)^n*hypergeom([-n,-n], [1], -3).

a(n) = (4/3)*(16/3)^n*hypergeom([n+1,n+1], [1], -1/3). (End)

EXAMPLE

G.f. = 1 + 2*x - 2*x^2 - 28*x^3 - 74*x^4 + 92*x^5 + 1324*x^6 + 3656*x^7 + ...

MAPLE

a := n -> 4^n*hypergeom([-n, 1+n], [1], 1/4);

seq(simplify(a(n)), n=0..26); # Peter Luschny, May 09 2016

MATHEMATICA

Table[ 2^(2n) LegendreP[ n, 1/2 ], {n, 12} ]

PROG

(PARI) {a(n) = 2^(2*n) * subst( pollegendre(n), x, 1/2)} /* Michael Somos, Dec 03 2001 */

(PARI) a(n) = polcoeff(((1 - x)*(1 + 3*x))^n, n); \\ Michel Marcus, Aug 16 2015

CROSSREFS

Cf. A116091, A098332, A126869.

Sequence in context: A193618 A246062 A178955 * A116091 A127262 A343172

Adjacent sequences:  A011997 A011998 A011999 * A012001 A012002 A012003

KEYWORD

sign,changed

AUTHOR

Wouter Meeussen

EXTENSIONS

More terms from Vladeta Jovovic, May 13 2003

STATUS

approved

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Last modified December 9 03:23 EST 2021. Contains 349625 sequences. (Running on oeis4.)