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%I M1849 #102 Jan 31 2023 11:20:11
%S 1,2,8,32,136,592,2624,11776,53344,243392,1116928,5149696,23835904,
%T 110690816,515483648,2406449152,11258054144,52767312896,247736643584,
%U 1164829376512,5484233814016,25852072517632,121997903495168
%N n*a(n) = 2*(2*n-1)*a(n-1) + 4*(n-1)*a(n-2) with a(0) = 1.
%C a(n) = number of Delannoy paths (A001850) from (0,0) to (n,n) in which every Northeast step is immediately preceded by an East step. - _David Callan_, Mar 14 2004
%C The Hankel transform (see A001906 for definition) of this sequence is A036442 : 1, 4, 32, 512, 16384, ... . - _Philippe Deléham_, Jul 03 2005
%C In general, 1/sqrt(1-4*r*x-4*r*x^2) has e.g.f. exp(2rx)BesselI(0,2r*sqrt((r+1)/r)x), a(n) = Sum_{k=0..n} C(2k,k)*C(k,n-k)*r^k, gives the central coefficient of (1+(2r)x+r(r+1)x^2) and is the (2r)-th binomial transform of 1/sqrt(1-8*C(n+1,2)x^2). - _Paul Barry_, Apr 28 2005
%C Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H and U steps can have two colors. - _N-E. Fahssi_, Feb 05 2008
%C Self-convolution of a(n)/2^n gives Pell numbers A000129(n+1). - _Vladimir Reshetnikov_, Oct 10 2016
%C This sequence gives the integer part of an integral approximation to Pi, and also appears in Frits Beukers's "A Rational Approach to Pi" (cf. Links, Example). Despite quality M ~ 0.9058... reported by Beukers, measurements between n = 10000 and 30000 lead to a contentious quality estimate, M ~ 0.79..., at the 99% confidence level. In "Searching for Apéry-Style Miracles" Doron Zeilberger Quotes that M = 0.79119792... and also gives a closed form. The same rational approximation to Pi also follows from time integration on a quartic Hamiltonian surface, 2*H=(q^2+p^2)*(1-4*q*(q-p)). - _Bradley Klee_, Jul 19 2018, updated Mar 17 2019
%C Diagonal of rational function 1/(1 - (x + y + x*y^2)). - _Gheorghe Coserea_, Aug 06 2018
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A006139/b006139.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from T. D. Noe)
%H Frits Beukers, <a href="https://dspace.library.uu.nl/handle/1874/26398">A rational approach to Pi</a>, Nieuw archief voor wiskunde 5/1 No. 4, December 2000, p. 377.
%H Dario Castellanos, <a href="http://www.fq.math.ca/Scanned/27-5/castellanos.pdf">A generalization of Binet's formula and some of its consequences</a>, Fib. Quart., 27 (1989), 424-438.
%H Maciej Dziemianczuk, <a href="http://arxiv.org/abs/1410.5747">On Directed Lattice Paths With Additional Vertical Steps</a>, arXiv:1410.5747 [math.CO], 2014.
%H Shalosh B. Ekhad and Doron Zeilberger, <a href="https://arxiv.org/abs/1405.4445">Searching for Apéry-Style Miracles [Using, Inter-Alia, the Amazing Almkvist-Zeilberger Algorithm]</a>, arXiv:1405.4445 [math.NT], 2014.
%H Bradley Klee, <a href="http://demonstrations.wolfram.com/ApproximatingPiWithTrigonometricPolynomialIntegrals/">Approximating Pi with Trigonometric-Polynomial Integrals</a>, Wolfram Demonstrations, July 27, 2018.
%H Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%F a(n) = Sum_{k=0..n} C(2*k, k)*C(k, n-k). - Detlef Pauly (dettodet(AT)yahoo.de), Nov 08 2001
%F G.f.: 1/(1-4x-4x^2)^(1/2); also, a(n) is the central coefficient of (1+2x+2x^2)^n. - _Paul D. Hanna_, Jun 01 2003
%F Inverse binomial transform of central Delannoy numbers A001850. - _David Callan_, Mar 14 2004
%F E.g.f.: exp(2*x)*BesselI(0, 2*sqrt(2)*x). - _Vladeta Jovovic_, Mar 21 2004
%F a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*C(2k,k)*2^(n-k). - _Paul Barry_, Sep 19 2006
%F a(n) ~ 2^(n - 3/4) * (1 + sqrt(2))^(n + 1/2) / sqrt(Pi*n). - _Vaclav Kotesovec_, Oct 05 2012, simplified Jan 31 2023
%F G.f.: 1/(1 - 2*x*(1+x)*Q(0)), where Q(k)= 1 + (4*k+1)*x*(1+x)/(k+1 - x*(1+x)*(2*k+2)*(4*k+3)/(2*x*(1+x)*(4*k+3)+(2*k+3)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 14 2013
%F a(n) = 2^n*hypergeom([-n/2, 1/2-n/2], [1], 2). - _Peter Luschny_, Sep 18 2014
%F 0 = a(n)*(+16*a(n+1) + 24*a(n+2) - 8*a(n+3)) + a(n+1)*(+8*a(n+1) + 16*a(n+2) - 6*a(n+3)) + a(n+2)*(-2*a(n+2) + a(n+3)) for all n in Z. - _Michael Somos_, Oct 13 2016
%F It appears that Pi/2 = Sum_{n >= 1} (-1)^(n-1)*4^n/(n*a(n-1)*a(n)). - _Peter Bala_, Feb 20 2017
%F G.f.: G(x) = (1/(2*Pi))*Integral_{y=0..2*Pi} 1/(1-x*(4*(sin(y)-cos(y))*sin(y)))*dy, also satisfies: (2+4*x)*G(x)-(1-4*x-4*x^2)*G'(x)=0. - _Bradley Klee_, Jul 19 2018
%e G.f. = 1 + 2*x + 8*x^2 + 32*x^3 + 136*x^4 + 592*x^5 + 2624*x^6 + 11776*x^7 + ...
%e J_3 = Integral_{y=0..Pi/4} 4*(4*(sin(y)-cos(y))*sin(y))^3*dy = 32*Pi - (304/3), |J_3| < 1. - _Bradley Klee_, Jul 19 2018
%p seq(add(binomial(2*k, k)*binomial(k, n-k), k=0..n), n=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 08 2001
%p A006139 := n -> 2^n*hypergeom([-n/2, 1/2-n/2], [1], 2):
%p seq(simplify(A006139(n)), n=0..29); # _Peter Luschny_, Sep 18 2014
%t Table[SeriesCoefficient[1/(1-4x-4x^2)^(1/2),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 05 2012 *)
%t Table[Abs[LegendreP[n, I]] 2^n, {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 22 2015 *)
%t Table[Sum[Binomial[2*k, k]*Binomial[k, n - k], {k,0,n}], {n,0,50}] (* _G. C. Greubel_, Feb 28 2017 *)
%t a[n_] := If[n == 0, 1, Coefficient[(1 + 2 x + 2 x^2)^n, x^n]] (* _Emanuele Munarini_, Aug 04 2017 *)
%t CoefficientList[Series[1/Sqrt[(-4 x^2 - 4 x + 1)], {x, 0, 24}], x] (* _Robert G. Wilson v_, Jul 28 2018 *)
%o (PARI) for(n=0,30,t=polcoeff((1+2*x+2*x^2)^n,n,x); print1(t","))
%o (PARI) for(n=0,25, print1(sum(k=0,n, binomial(2*k,k)*binomial(k,n-k)), ", ")) \\ _G. C. Greubel_, Feb 28 2017
%o (PARI) {a(n) = (-2*I)^n * pollegendre(n, I)}; /* _Michael Somos_, Aug 04 2018 */
%o (Maxima) a(n) := coeff(expand((1+2*x+2*x^2)^n),x,n);
%o makelist(a(n),n,0,12); /* _Emanuele Munarini_, Aug 04 2017 */
%o (GAP) a:=[1,2];; for n in [3..25] do a[n]:=1/(n-1)*(2*(2*n-3)*a[n-1]+4*(n-2)*a[n-2]); od; a; # _Muniru A Asiru_, Aug 06 2018
%Y Cf. A001850, A002426, A036442, A084600-A084606, A084608-A084615.
%Y Cf. A106258, A106259, A106260, A106261.
%Y First column of A110446. A higher-quality Pi approximation: A123178.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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