OFFSET
0,2
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.
FORMULA
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
MAPLE
A005430 := n -> n*binomial(2*n, n);
MATHEMATICA
Table[n*Binomial[2n, n], {n, 0, 30}] (* Harvey P. Dale, May 29 2015 *)
PROG
(PARI) a(n)=-(-1)^n*real(polcoeff(serlaplace(x^2*besselh1(1, 2*x)), 2*n)) \\ Ralf Stephan
(Magma) [n*Binomial(2*n, n): n in [0..30]]; // G. C. Greubel, Dec 09 2018
(Sage) [n*binomial(2*n, n) for n in range(30)] # G. C. Greubel, Dec 09 2018
(GAP) List([0..30], n-> n*Binomial(2*n, n)); # G. C. Greubel, Dec 09 2018
CROSSREFS
KEYWORD
nonn,easy,nice,changed
AUTHOR
EXTENSIONS
More terms from James A. Sellers, May 01 2000
STATUS
approved