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A082590 Expansion of 1/((1 - 2*x)*sqrt(1 - 4*x)).

%I A082590 #72 May 23 2025 01:09:59
%S A082590 1,4,14,48,166,584,2092,7616,28102,104824,394404,1494240,5692636,
%T A082590 21785872,83688344,322494208,1246068806,4825743832,18726622964,
%U A082590 72798509728,283443548276,1105144970992,4314388905704,16862208539008,65972020761116,258354647959984,1012627828868072
%N A082590 Expansion of 1/((1 - 2*x)*sqrt(1 - 4*x)).
%C A082590 Row sums of A068555 and A112336. - _Paul Barry_, Sep 04 2005
%C A082590 Hankel transform is 2^n*(-1)^C(n+1,2) (A120617). - _Paul Barry_, Apr 26 2009
%C A082590 Number of n-lettered words in the alphabet {1, 2, 3, 4} with as many occurrences of the substring (consecutive subword) [1, 2] as of [1, 3]. - _N. J. A. Sloane_, Apr 08 2012
%H A082590 Vincenzo Librandi, <a href="/A082590/b082590.txt">Table of n, a(n) for n = 0..300</a>
%H A082590 Shalosh B. Ekhad and Doron Zeilberger, <a href="http://arxiv.org/abs/1112.6207">Automatic Solution of Richard Stanley's Amer. Math. Monthly Problem #11610 and ANY Problem of That Type</a>, arXiv:1112.6207 [math.CO], 2011. See subpages for rigorous derivations of the g.f., the recurrence, asymptotics for this sequence.
%H A082590 Alejandro Erickson and Frank Ruskey, <a href="http://arxiv.org/abs/1304.0070">Enumerating maximal tatami mat coverings of square grids with v vertical dominoes</a>, arXiv:1304.0070 [math.CO], 2013.
%H A082590 Y. Kamiyama, <a href="http://arxiv.org/abs/1507.03161">On the middle dimensional homology classes of equilateral polygon spaces</a>, arXiv:1507.03161 [math.AT], 2015.
%F A082590 a(n) = 2^n*JacobiP(n, 1/2, -1-n, 3).
%F A082590 A034430(n) = (n!/2^n)*a(n). A076729(n) = n!*a(n).
%F A082590 a(n) = Sum_{k=0..n+1} binomial(2*n+2, k) * sin((n - k + 1)*Pi/2). - _Paul Barry_, Nov 02 2004
%F A082590 From _Paul Barry_, Sep 04 2005: (Start)
%F A082590 a(n) = Sum_{k=0..n} 2^(n-k)*binomial(2*k, k).
%F A082590 a(n) = Sum_{k=0..n} (2*k)! * (2*(n-k))!/(n!*k!*(n-k)!). (End)
%F A082590 a(n) = Sum_{k=0..n} C(2*n, n)*C(n, k)/C(2*n, 2*k). - _Paul Barry_, Mar 18 2007
%F A082590 G.f.: 1/(1 - 4*x + 2*x^2/(1 + x^2/(1 - 4*x + x^2/(1 + x^2/(1 - 4*x + x^2/(1 + ... (continued fraction). - _Paul Barry_, Apr 26 2009
%F A082590 D-finite with recurrence: n*a(n) + 2*(-3*n+1)*a(n-1) + 4*(2*n-1)*a(n-2) = 0. - _R. J. Mathar_, Dec 03 2012
%F A082590 a(n) ~ 2^(2*n + 1)/sqrt(Pi*n). - _Vaclav Kotesovec_, Aug 15 2013
%F A082590 a(n) = 2^(n + 1)*Pochhammer(1/2, n+1)*hyper2F1([1/2,-n], [3/2], -1)/n!. - _Peter Luschny_, Aug 02 2014
%F A082590 a(n) - 2*a(n-1) = A000984(n). - _R. J. Mathar_, Apr 24 2024
%F A082590 a(n) = 2^n*JacobiP(n, 1/2, -1 - n, 3). - _Peter Luschny_, Jan 22 2025
%p A082590 A082590 := proc(n)
%p A082590     coeftayl( 1/(1-2*x)/sqrt(1-4*x),x=0,n) ;
%p A082590 end proc: # _R. J. Mathar_, Nov 06 2013
%p A082590 A082590 := n -> 2^n*JacobiP(n, 1/2, -1 - n, 3):
%p A082590 seq(simplify(A082590(n)), n = 0..26);  # _Peter Luschny_, Jan 22 2025
%t A082590 CoefficientList[ Series[ 1/((1 - 2*x)*Sqrt[1 - 4*x]), {x, 0, 25}], x] (* _Jean-François Alcover_, Mar 26 2013 *)
%t A082590 Table[2^(n) JacobiP[n, 1/2, -1-n, 3], {n, 0, 30}] (* _Vincenzo Librandi_, May 26 2013 *)
%Y A082590 Bisection of A226302.
%Y A082590 Cf. A034430, A068555, A076729, A112336.
%K A082590 nonn
%O A082590 0,2
%A A082590 _Vladeta Jovovic_, May 13 2003