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A098270 a(n) = 2^n*P_n(5), 2^n times the Legendre polynomial of order n at 5. 3
1, 10, 148, 2440, 42256, 752800, 13660480, 251113600, 4660568320, 87140108800, 1638884021248, 30970912737280, 587599919386624, 11185644310405120, 213540626285805568, 4086692369433395200, 78378887309200261120 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Central coefficients of (1+10x+24x^2)^n. 2^n*LegendreP(n,k) yields the central coefficients of (1+2kx+(k^2-1)x^2)^n, with g.f. 1/sqrt(1-4kx+4x^2).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Eric Weisstein's World of Mathematics, Legendre Polynomial.

FORMULA

G.f.: 1/sqrt(1-20x+4x^2).

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

Conjecture: n*a(n) +10*(1-2*n)*a(n-1) +4*(n-1)*a(n-2)=0. - R. J. Mathar, Sep 26 2012

a(n) ~ sqrt(72+30*sqrt(6))*(10+4*sqrt(6))^n/(12*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012

a(n) = A059473(n,n). - Alois P. Heinz, Oct 05 2017

MATHEMATICA

Table[SeriesCoefficient[1/Sqrt[1-20*x+4*x^2], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *)

PROG

(PARI) a(n)=pollegendre(n, 5)<<n \\ Charles R Greathouse IV, Oct 25 2011

(Sage)

def A098270(n): return 2^n*gen_legendre_P(n, 0, 5)

[A098270(n) for n in (0..16)] # Peter Luschny, Oct 14 2012

CROSSREFS

Cf. A069835, A084773.

Cf. A059473.

Sequence in context: A095889 A097638 A178084 * A262738 A271467 A212471

Adjacent sequences:  A098267 A098268 A098269 * A098271 A098272 A098273

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Sep 01 2004

STATUS

approved

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Last modified June 19 12:57 EDT 2019. Contains 324222 sequences. (Running on oeis4.)