%I M2028 #113 Jul 11 2023 12:21:30

%S 0,2,12,60,280,1260,5544,24024,102960,437580,1847560,7759752,32449872,

%T 135207800,561632400,2326762800,9617286240,39671305740,163352435400,

%U 671560012200,2756930576400,11303415363240,46290177201840,189368906734800,773942488394400

%N Apéry numbers: n*C(2*n,n).

%C Sum_{n >= 1} 1/a(n) = Pi*sqrt(3)/9 - _Benoit Cloitre_, Apr 07 2002

%C Appears as diagonal in A003506. - _Zerinvary Lajos_, Apr 12 2006

%C The aerated sequence 1,0,2,0,12,0,60,0,... has e.g.f. 1+x*Bessel_I(1,2x). - _Paul Barry_, Mar 29 2010

%C Conjecture: the terms of the inverse binomial transform are 2*A132894(n). - _R. J. Mathar_, Oct 21 2012

%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 78, (3.5.25).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A005430/b005430.txt">Table of n, a(n) for n = 0..200</a>

%H L. Alonso and E. M. Reingold, <a href="https://hal.inria.fr/hal-00926106">Analysis of Boyer and Moore's MJRTY Algorithm</a>, 2012.

%H T. Amdeberhan and Henri Cohen, <a href="https://mathoverflow.net/q/272258">Bernoulli sum meets golden number</a>, MathOverflow, version of 2017-06-15.

%H Libor Caha and Daniel Nagaj, <a href="https://arxiv.org/abs/1805.07168">The pair-flip model: a very entangled translationally invariant spin chain</a>, arXiv:1805.07168 [quant-ph], 2018.

%H Benjamin Ruoyu Kan, <a href="https://dash.harvard.edu/handle/1/37376406">Polynomial Approximations for Quantum Hamiltonian Complexity</a>, Bachelor's thesis, Harvard Univ., 2023.

%H H. J. H. Tuenter, <a href="https://arxiv.org/abs/math/0606080">Walking into an absolute sum</a>, arXiv:math/0606080 [math.NT], 2006. Published version on <a href="http://www.fq.math.ca/Scanned/40-2/tuenter.pdf">Walking into an absolute sum</a>, The Fibonacci Quarterly, 40(2):175-180, May 2002.

%H A. J. van der Poorten, <a href="http://dx.doi.org/10.1007/BF03028234">A proof that Euler missed...Apery's proof of the irrationality of zeta(3)</a>, Math. Intelligencer 1 (1978/1979), 195-203.

%H I. J. Zucker, <a href="http://dx.doi.org/10.1016/0022-314X(85)90019-8">On the series Sum(k>=1) C(2k,k)^(-1)*k^(-n) and related sums</a>, J. Number Theory 20 (1985), no. 1, 92-102.

%H Wadim Zudilin, <a href="http://arXiv.org/abs/math/0202159">An elementary proof of Apery's theorem</a>, arXiv:math/0202159 [math.NT], 2002.

%F a(n) = A002011(n-1)/2 = 2 * A002457(n-1).

%F G.f.: 2*x/sqrt((1-4*x)^3). - _Marco A. Cisneros Guevara_, Jul 25 2011

%F E.g.f.: a(n) = n!* [x^n] exp(2*x)*2*x*(BesselI(0, 2*x)+BesselI(1, 2*x)). - _Peter Luschny_, Aug 25 2012

%F D-finite with recurrence (-n+1)*a(n) + 2*(2*n-1)*a(n-1) = 0. - _R. J. Mathar_, Dec 03 2012

%F G.f.: 2*x*(1-4*x)^(-3/2) = -G(0)/2 where G(k) = 1 - (2*k+1)/(1 - 2*x/(2*x - (k+1)/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Dec 06 2012

%F a(n-1) = Sum_{k=0..floor(n/2)} k*C(n,k)*C(n-k,k)*2^(n-2*k). - _Robert FERREOL_, Aug 29 2015

%F From _Ilya Gutkovskiy_, Jan 17 2017: (Start)

%F a(n) ~ 4^n*sqrt(n)/sqrt(Pi).

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(phi)/sqrt(5) = A086466, where phi is the golden ratio. (End)

%F 1/a(n) = (-1)^n*Sum_{j=0..n-1} binomial(n-1,j)*Bernoulli(j+n)/(j+n) for n >= 1. See the Amdeberhan & Cohen link. - _Peter Luschny_, Jun 20 2017

%F 1/a(n) = Sum_{k=0..n} (-1)^(k+1)*binomial(n,k)*HarmonicNumber(n+k) for n >= 1. - _Peter Luschny_, Aug 15 2017

%p A005430 := n -> n*binomial(2*n, n);

%t Table[n*Binomial[2n,n],{n,0,30}] (* _Harvey P. Dale_, May 29 2015 *)

%o (PARI) a(n)=-(-1)^n*real(polcoeff(serlaplace(x^2*besselh1(1,2*x)),2*n)) \\ _Ralf Stephan_

%o (Magma) [n*Binomial(2*n,n): n in [0..30]]; // _G. C. Greubel_, Dec 09 2018

%o (Sage) [n*binomial(2*n,n) for n in range(30)] # _G. C. Greubel_, Dec 09 2018

%o (GAP) List([0..30], n-> n*Binomial(2*n,n)); # _G. C. Greubel_, Dec 09 2018

%Y Cf. A002736, A005258, A005259, A005429. 1/beta(n, n+1) in A061928.

%Y Cf. A001803, A003506.

%K nonn,easy,nice

%O 0,2

%A _Simon Plouffe_

%E More terms from _James A. Sellers_, May 01 2000