A005430 Apéry numbers: n*C(2*n,n).

0, 2, 12, 60, 280, 1260, 5544, 24024, 102960, 437580, 1847560, 7759752, 32449872, 135207800, 561632400, 2326762800, 9617286240, 39671305740, 163352435400, 671560012200, 2756930576400, 11303415363240, 46290177201840, 189368906734800, 773942488394400

Offset: 0

Author

Simon Plouffe

Keywords

• nonn
• easy
• nice

Comments

Appears as diagonal in A003506. - Zerinvary Lajos, Apr 12 2006

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

Conjecture: the terms of the inverse binomial transform are 2*A132894(n). - R. J. Mathar, Oct 21 2012

References

• Frank Harary and Edgar M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 78, (3.5.25).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..200
• Kunle Adegoke, Robert Frontczak, and Taras Goy, Fibonacci-Catalan Series, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 22 (2022), #A110.
• Laurent Alonso and Edward M. Reingold, Analysis of Boyer and Moore's MJRTY Algorithm, 2012.
• T. Amdeberhan and Henri Cohen, Bernoulli sum meets golden number, MathOverflow, version of 2017-06-15.
• Libor Caha and Daniel Nagaj, The pair-flip model: a very entangled translationally invariant spin chain, arXiv:1805.07168 [quant-ph], 2018.
• Benjamin Ruoyu Kan, Polynomial Approximations for Quantum Hamiltonian Complexity, Bachelor's thesis, Harvard Univ., 2023.
• Hans J. H. Tuenter, Walking into an absolute sum, arXiv:math/0606080 [math.NT], 2006. Published version on Walking into an absolute sum, The Fibonacci Quarterly, 40(2):175-180, May 2002.
• Alfred J. van der Poorten, A proof that Euler missed ... Apery's proof of the irrationality of zeta(3), Math. Intelligencer 1 (1978/1979), 195-203.
• I. J. Zucker, On the series Sum(k>=1) C(2k,k)^(-1)*k^(-n) and related sums, J. Number Theory, Vol. 20, No. 1 (1985), 92-102.
• Wadim Zudilin, An elementary proof of Apery's theorem, arXiv:math/0202159 [math.NT], 2002.

Crossrefs

Cf. A002011, A002457, A002736, A005258, A005259, A005429, 1/beta(n, n+1) in A061928.

Cf. A001803, A003506.

Programs

GAP

List([0..30], n-> n*Binomial(2*n,n)); # G. C. Greubel, Dec 09 2018

Magma

[n*Binomial(2*n,n): n in [0..30]]; // G. C. Greubel, Dec 09 2018

Maple

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

Mathematica

Table[n*Binomial[2n,n],{n,0,30}] (* Harvey P. Dale, May 29 2015 *)

PARI

a(n)=-(-1)^n*real(polcoeff(serlaplace(x^2*besselh1(1,2*x)),2*n)) \\ Ralf Stephan

Sage

[n*binomial(2*n,n) for n in range(30)] # G. C. Greubel, Dec 09 2018

Formula

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

Sum_{n >= 1} 1/a(n) = Pi*sqrt(3)/9. - Benoit Cloitre, Apr 07 2002

G.f.: 2*x/sqrt((1-4*x)^3). - Marco A. Cisneros Guevara, Jul 25 2011

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

D-finite with recurrence (-n+1)*a(n) + 2*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012

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

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

From Ilya Gutkovskiy, Jan 17 2017: (Start)

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

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

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

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

Sum_{n>=1} x^n/a(n) = 2*sqrt(x/(4-x))*arcsin(sqrt(x)/2), for abs(x) < 4 (Adegoke et al., 2022, section 6, p. 11). - Amiram Eldar, Dec 07 2024

Extensions

More terms from James Sellers, May 01 2000