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A284016 a(-1)=-1; a(n) = 2*A000108(n) for n >= 0. 3
-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 (list; graph; refs; listen; history; text; internal format)
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

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)

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

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

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, A068875, A228403, A262543.

Sequence in context: A078801 A309159 A002420 * A112556 A254400 A054100

Adjacent sequences:  A284013 A284014 A284015 * A284017 A284018 A284019

KEYWORD

sign

AUTHOR

Ralf Steiner, Mar 28 2017

STATUS

approved

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Last modified January 20 14:24 EST 2020. Contains 331094 sequences. (Running on oeis4.)