A284016 a(-1)=-1; a(n) = 2*A000108(n) for n >= 0.

-1, 2, 2, 4, 10, 28, 84, 264, 858, 2860, 9724, 33592, 117572, 416024, 1485800, 5348880, 19389690, 70715340, 259289580, 955277400, 3534526380, 13128240840, 48932534040, 182965127280, 686119227300, 2579808294648, 9723892802904, 36734706144304, 139067101832008, 527495903500720, 2004484433302736

Offset

-1,2

Comments

There exists a set of Ramanujan-Sato series using this sequence.

a(n-1) = binomial(2n,n)/(2n-1) is the number of walks on a line that start and end at origin after 2n steps, not touching origin at intermediate stages. - Robert FERREOL, Aug 24 2019

From Bernard Schott, Apr 18 2020: (Start)

If E_n is the set of sequences with 2n digits composed with n digits 0 and n digits 1, then #E_n = binomial(2n,n). Now, if F_n is the subset of E_n composed of sequences where number of 0's = number of 1's for the first time at index 2n, then #F_n = a(n-1) = binomial(2n,n)/(2n-1) [see reference for proof].

Example: For n = 2, E_2 = {(0,0,1,1), (0,1,0,1), (1,0,1,0), (1,1,0,0), (0,1,1,0), (1,0,0,1)} with #E_2 = 6, but, F_2 = {(0,0,1,1), (1,1,0,0)} and #F_2 = a(1) = 2. (End)

References

Patrice Tauvel, Exercices d'Algèbre Générale et d'Arithmétique, Dunod, 2004, Exercice 11, p. 296.

Links

G. C. Greubel, Table of n, a(n) for n = -1..1000

Wikipedia, Ramanujan-Sato series.

Formula

a(n) = -A002420(n+1).

a(n) = (2*4^n*Gamma(n+1/2))/(sqrt(Pi)*Gamma(n+2)) for n >= -1. - Ralf Steiner, Apr 02 2017

a(n) = binomial(2*n+2, n+1) / (2*n+1) = 4*binomial(2*n, n) - binomial(2*n+2, n+1) for all n in Z. - Michael Somos, Jan 26 2018

a(n) = A228403(n) for n > 1 (essentially twice the Catalan numbers). - Georg Fischer, Oct 23 2018

From Stefano Spezia, Aug 24 2019: (Start)

G.f. for n >= 0: (1 - sqrt(1 - 4*x))/x.

E.g.f. for n >= 0: 2*(exp(2*x))*(I_{0}(2*x) - I_{1}(2*x)) where I_{k}(x) is the modified Bessel function of the first kind. (End)

Sum_{n>=-1} 1/a(n) = 2*Pi/(9*sqrt(3)). - Amiram Eldar, Jan 18 2025

Example

The a(3)=10 8-step walks starting from and ending at the origin are [0, -1, -2, -3, -4, -3, -2, -1, 0], [0, -1, -2, -3, -2, -3, -2, -1, 0], [0, -1, -2, -3, -2, -1, -2, -1, 0], [0, -1, -2, -1, -2, -3, -2, -1, 0], [0, -1, -2, -1, -2, -1, -2, -1, 0], [0, 1, 2, 1, 2, 1, 2, 1, 0], [0, 1, 2, 1, 2, 3, 2, 1, 0], [0, 1, 2, 3, 2, 1, 2, 1, 0], [0, 1, 2, 3, 2, 3, 2, 1, 0], [0, 1, 2, 3, 4, 3, 2, 1, 0]. - Robert FERREOL, Aug 24 2019

Maple

seq(binomial(2*n+2, n+1) / (2*n+1), n=-1..20); # Robert FERREOL, Aug 24 2019

Mathematica

Prepend[Table[(2*CatalanNumber[n]), {n, 0, 20}], -1]

Prog

(PARI) for(n=-1, 25, print1(round((2*4^n*gamma(n+1/2))/(sqrt(Pi)*gamma(n+2))), ", ")) \\ G. C. Greubel, Apr 11 2017
(PARI) {a(n) = binomial(2*n+2, n+1) / (2*n+1)}; /* Michael Somos, Jan 26 2018 */
(Magma) [Binomial(2*n+2, n+1) / (2*n+1): n in [-1..30]]; // Vincenzo Librandi, Jan 27 2018

Crossrefs

Cf. A000108, A002420, A228403.

Essentially the same as A262543 and A068875.

Sequence in context: A078801 A309159 A002420 * A112556 A254400 A054100

Adjacent sequences: A284013 A284014 A284015 * A284017 A284018 A284019

Keyword

sign,changed

Author

Ralf Steiner, Mar 28 2017

Status

approved