The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A172060

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A172060

The number of returns to the origin in all possible one-dimensional walks of length 2n.
(history; published version)

#23 by Charles R Greathouse IV at Thu Sep 08 08:45:50 EDT 2022

PROG

(MAGMAMagma) [Factorial(2*n+1)/Factorial(n)^2 - 4^n : n in [0..30]]; // Wesley Ivan Hurt, Mar 24 2015

(MAGMAMagma) [0] cat [n le 1 select 2 else 4*Self(n-1)+ Binomial(2*n, n): n in [1..30]]; // Vincenzo Librandi, Mar 25 2015

Discussion

Thu Sep 08

08:45

OEIS Server: https://oeis.org/edit/global/2944

#22 by Jon E. Schoenfield at Sun Jan 03 00:37:03 EST 2021

STATUS

editing

approved

#21 by Jon E. Schoenfield at Sun Jan 03 00:37:01 EST 2021

REFERENCES

W. Feller, W., "An Introduction to Probability Theory and its Applications", , Vol 1, 3rd ed. New York: Wiley, pp. 67-97, 1968.

STATUS

approved

editing

#20 by R. J. Mathar at Wed Jan 03 07:31:15 EST 2018

STATUS

editing

approved

#19 by R. J. Mathar at Wed Jan 03 07:27:57 EST 2018

FORMULA

a(n) = 2*A000531(n). - R. J. Mathar, Jan 03 2018

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approved

editing

#18 by Bruno Berselli at Wed Mar 25 06:04:17 EDT 2015

STATUS

proposed

approved

#17 by Vincenzo Librandi at Wed Mar 25 01:49:40 EDT 2015

STATUS

editing

proposed

#16 by Vincenzo Librandi at Wed Mar 25 01:49:30 EDT 2015

MATHEMATICA

CoefficientList[Series[(1 - (1 - 4 x)^(1/2)) / (1 - 4 x)^(3/2), {x, 0, 33}], x] (* Vincenzo Librandi, Mar 25 2015 *)

PROG

(MAGMA) [0] cat [n le 1 select 2 else 4*Self(n-1)+ Binomial(2*n, n): n in [1..30]]; // Vincenzo Librandi, Mar 25 2015

STATUS

proposed

editing

#15 by Jon E. Schoenfield at Tue Mar 24 23:26:17 EDT 2015

STATUS

editing

proposed

#14 by Jon E. Schoenfield at Tue Mar 24 23:26:09 EDT 2015

NAME

The number of returns to the origin in all possible one -dimensional walks of length 2n.

COMMENTS

a(n)/4^n is the expected number of times a gambler will return to his break -even point while making 2n equal wagers on the outcome of a fair coin toss. Note the surprisingly low and slow growth of this expectation.

STATUS

proposed

editing