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A098484

Expansion of 1/sqrt((1-x)^2-12x^4).
(history; published version)

#12 by N. J. A. Sloane at Thu Jan 30 21:29:15 EST 2020

FORMULA

D-finite with recurrence: n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + 12*(n-2)*a(n-4). - Vaclav Kotesovec, Jun 23 2014

Discussion

Thu Jan 30

21:29

OEIS Server: https://oeis.org/edit/global/2847

#11 by R. J. Mathar at Mon Jan 20 05:14:30 EST 2020

STATUS

editing

approved

#10 by R. J. Mathar at Mon Jan 20 05:14:26 EST 2020

FORMULA

RecurrenceD-finite: n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + 12*(n-2)*a(n-4). - Vaclav Kotesovec, Jun 23 2014

STATUS

approved

editing

#9 by Vaclav Kotesovec at Mon Jun 23 15:35:28 EDT 2014

STATUS

proposed

approved

#8 by Vincenzo Librandi at Mon Jun 23 13:46:05 EDT 2014

STATUS

editing

proposed

#7 by Vincenzo Librandi at Mon Jun 23 13:45:59 EDT 2014

LINKS

Vincenzo Librandi, <a href="/A098484/b098484.txt">Table of n, a(n) for n = 0..200</a>

STATUS

approved

editing

#6 by Vaclav Kotesovec at Mon Jun 23 08:12:08 EDT 2014

STATUS

editing

approved

#5 by Vaclav Kotesovec at Mon Jun 23 08:11:47 EDT 2014

COMMENTS

1/sqrt((1-x)^2-4rx^4) expands to sum{k=0..floor(n/2), binomial(n-2k,k)binomial(n-3k,k)r^k}.

#4 by Vaclav Kotesovec at Mon Jun 23 08:10:54 EDT 2014

FORMULA

a(n)=sum{k=0..floor(n/2), binomial(n-2k, k)binomial(n-3k, k)3^k}.

Recurrence: n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + 12*(n-2)*a(n-4). - Vaclav Kotesovec, Jun 23 2014

a(n) ~ sqrt(3) * (1+sqrt(1+8*sqrt(3)))^n / (sqrt(49+10*sqrt(3)-sqrt(397+884*sqrt(3))) * sqrt(Pi*n) * 2^(n-1)). - Vaclav Kotesovec, Jun 23 2014

MATHEMATICA

CoefficientList[Series[1/Sqrt[(1-x)^2-12*x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 23 2014 *)

STATUS

approved

editing

#3 by Russ Cox at Fri Mar 30 18:58:59 EDT 2012

AUTHOR

_Paul Barry (pbarry(AT)wit.ie), _, Sep 10 2004

Discussion

Fri Mar 30

18:58

OEIS Server: https://oeis.org/edit/global/287