|
| |
|
|
A284016
|
|
a(-1)=-1; a(n) = 2*A000108(n) for n >= 0.
|
|
6
|
|
|
|
-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
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
|
P. Tauvel, Exercices d'Algèbre Générale et d'Arithmétique, Dunod, 2004, Exercice 11 p. 296.
|
|
|
LINKS
|
|
|
|
FORMULA
|
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
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
|
|
|
|
KEYWORD
|
sign
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
