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A212199 Expansion of (1 + 3*x + 4*x^2 - sqrt(1 - 2*x - 7*x^2))/(4 + 8*x). 1

%I #37 Oct 19 2020 07:40:42

%S 0,1,0,1,1,5,11,39,113,377,1207,4043,13509,45957,157171,542671,

%T 1884665,6586993,23137647,81662355,289414157,1029598333,3675337963,

%U 13160833623,47261437761,170164260713,614154154791,2221545593179,8052506141653,29244341625077,106397352342243,387745600670175,1415284544031241,5173441096267489,18937206005320415,69409364862108451

%N Expansion of (1 + 3*x + 4*x^2 - sqrt(1 - 2*x - 7*x^2))/(4 + 8*x).

%H Vincenzo Librandi, <a href="/A212199/b212199.txt">Table of n, a(n) for n = 0..200</a>

%H Sergey Kitaev, Pavel Salimov, Christopher Severs, and Henning Ulfarsson, <a href="http://arxiv.org/abs/1202.1790">Restricted rooted non-separable planar maps</a>, arXiv:1202.1790 [math.CO], 2012.

%H S. Kitaev, P. Salimov, C. Severs and H. Ulfarsson, <a href="http://staff.ru.is/henningu/papers/maps/maps.pdf">Restricted non-separable planar maps and some pattern avoiding permutations</a>, preprint 2012. [_N. J. A. Sloane_, Jan 01 2013]

%F a(n) ~ sqrt(44-31*sqrt(2))*(1+2*sqrt(2))^n/(4*n^(3/2)*sqrt(Pi)). - _Vaclav Kotesovec_, Jun 29 2013

%F D-finite with recurrence: n*a(n) +3*a(n-1) -(11*n-27)*a(n-2) -14*(n-3)*a(n-3) = 0 for n>4. - _Bruno Berselli_, Jul 18 2013

%F Let T(n, k) = 2^k*binomial(n-k, k)*hypergeom([-k, k-n-1], [2], 1) then

%F a(n) = Sum_{k=0..(n-3)/2} T(n-3, k) if n != 1. - _Peter Luschny_, Oct 19 2020

%p T := (n, k) -> simplify(2^k*binomial(n-k, k)*hypergeom([-k, k-n-1], [2], 1)):

%p [0, 1, 0, seq(add(T(n, k), k=0..floor(n/2)), n=0..32)]; # _Peter Luschny_, Oct 19 2020

%t CoefficientList[Series[(1 + 3 x + 4 x^2 - Sqrt[1 - 2 x - 7 x^2]) / (4 + 8 x), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 18 2013 *)

%o (PARI) x='x+O('x^50); concat([0], Vec((1+3*x+4*x^2-sqrt(1-2*x-7*x^2))/(4+8*x))) \\ _G. C. Greubel_, Mar 30 2017

%K nonn

%O 0,6

%A _N. J. A. Sloane_, May 11 2012

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