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A181784 Numerators of a series sum related to a game of chance. 1
1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3782992, 32389076, 275617830, 2350749914, 20140518790, 173429992350, 1500850805160, 14550277251918, 133009333771170, 1198324107797254 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Consider a 1-dimensional random walk from 0 with equal-probability steps of Pi and -1.  One way to compute the probability of eventually walking below 0 is as the sum over n of the probabilities of becoming negative after a walk with exactly n steps of Pi (n >=0) and max(ceil(n*pi),1) steps of -1.  The total number of walks of such length for a given n is 2^(n+max(ceil(n*pi),1)), or 2^(n+A004084(n)) (n>=1), forming a sequence of denominators, and this sequence gives the numerators, the number of possible sequences of length (n+max(ceil(n*pi),1)) drawn from {Pi, -1} such that no partial sum except the total sum is < 0.

See the Munafo web page for complete description.

a(n) diverges from A002293 because pi is not exactly 3.

LINKS

Table of n, a(n) for n=0..18.

R. Munafo, Related to a Game of Chance

"My Math Forum" discussion thread, I give, duz... what is it?

"duz" blog entry, Random Walking

EXAMPLE

Numerators of series sum 1/2 + 1/32 + 4/512 + 22/8192 + 140/131072 + ...

CROSSREFS

Sequence in context: A216712 A240586 A002293 * A003287 A077056 A227404

Adjacent sequences:  A181781 A181782 A181783 * A181785 A181786 A181787

KEYWORD

nonn,frac

AUTHOR

Robert Munafo, Dec 21 2010

EXTENSIONS

a(18) from Robert Munafo, Dec 22 2010

STATUS

approved

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Last modified November 23 21:46 EST 2017. Contains 295141 sequences.