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A098483 Expansion of 1/sqrt((1-x)^2-8x^4). 3

%I

%S 1,1,1,1,5,13,25,41,85,205,473,985,2021,4365,9785,21673,46965,101581,

%T 222745,492665,1087237,2388749,5251065,11587529,25633045,56697933,

%U 125345113,277283353,614212133,1361824525,3020426681,6700678377

%N Expansion of 1/sqrt((1-x)^2-8x^4).

%C 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}

%H Vincenzo Librandi, <a href="/A098483/b098483.txt">Table of n, a(n) for n = 0..200</a>

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

%F D-finite with recurrence: n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + 8*(n-2)*a(n-4). - _Vaclav Kotesovec_, Jun 23 2014

%F a(n) ~ (1+sqrt(1+8*sqrt(2)))^n / (sqrt(33+10*sqrt(2)-sqrt(265+596*sqrt(2))) * sqrt(Pi*n) * 2^(n-3/2)). - _Vaclav Kotesovec_, Jun 23 2014

%t CoefficientList[Series[1/Sqrt[(1-x)^2-8*x^4], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jun 23 2014 *)

%o (PARI) a(n) = sum(k=0, n\2, binomial(n-2*k, k)*binomial(n-3*k, k)*2^k) \\ _Michel Marcus_, Jul 24 2013

%Y Cf. A098480, A098482, A098484.

%K easy,nonn

%O 0,5

%A _Paul Barry_, Sep 10 2004

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Last modified March 20 04:42 EDT 2023. Contains 361358 sequences. (Running on oeis4.)