A108886 Let T(m,p) be the value of the following game: there are m "minus" balls and p "plus" balls in an urn, for a total of n=m+p balls. You may draw balls from the urn one at a time at random and without replacement until you decide to stop drawing. Each minus ball drawn costs you $1 and each plus ball drawn gets you $1. Sequence gives triangle of denominators of T(n-p,p), 0 <= p <= n, read by rows.

1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 1, 5, 2, 5, 1, 1, 1, 1, 20, 5, 6, 1, 1, 1, 1, 35, 35, 3, 7, 1, 1, 1, 1, 1, 1, 28, 7, 8, 1, 1, 1, 1, 1, 9, 63, 42, 4, 9, 1, 1, 1, 1, 1, 15, 42, 35, 120, 9, 10, 1, 1, 1, 1, 1, 1, 231, 11, 66, 55, 5, 11, 1, 1, 1, 1, 1, 1, 396, 231, 72, 45, 55, 11, 12, 1, 1, 1

Offset

0,5

References

L. A. Shepp, Stochastic Processes [Course], Statistics Dept., Rutgers University, 2004.

Links

Table of n, a(n) for n=0..92.

Formula

T(m, 0)=0, T(0, p)=p; T(m, p) = max{0, (m/(m+p))*(-1+T(m-1, p))+(p/(m+p))*(1+T(m, p-1))}.

Example

Triangle of values T(n-p,p), 0 <= p <= n, begins:
[0]
[0, 1]
[0, 1/2, 2]
[0, 0, 4/3, 3]
[0, 0, 2/3, 9/4, 4]
[0, 0, 1/5, 3/2, 16/5, 5]
[0, 0, 0, 17/20, 12/5, 25/6, 6]
[0, 0, 0, 12/35, 58/35, 10/3, 36/7, 7]
[0, 0, 0, 0, 1, 71/28, 30/7, 49/8, 8]

Maple

M:=60; for m from 0 to M do T(m, 0):=0; od: for p from 0 to M do T(0, p):=p; od: for n from 1 to M do for m from 1 to n-1 do p:=n-m; t1:=(m/(m+p))*(-1+T(m-1, p))+(p/(m+p))*(1+T(m, p-1)); T(m, p):=max(0, t1); od: od:

Mathematica

M = 60; Clear[T]; For[m = 0, m <= M, m++, T[m, 0] = 0]; For[p = 0, p <= M, p++, T[0, p] = p]; For[n = 1, n <= M, n++, For[m = 1, m <= n-1, m++, p = n-m; t1 = (m/(m+p))*(-1+T[m-1, p]) + (p/(m+p))*(1+T[m, p-1]); T[m, p] = Max[0, t1]]]; Table[T[n-p, p] // Denominator, {n, 0, 13}, {p, 0, n}] // Flatten (* Jean-François Alcover, Jan 09 2014, translated from Maple *)

Crossrefs

Cf. A108885. Sequence T(m, m) is A108883/A108884.

Sequence in context: A115622 A294892 A371211 * A140886 A001492 A347049

Adjacent sequences: A108883 A108884 A108885 * A108887 A108888 A108889

Keyword

nonn,tabl,frac

Author

N. J. A. Sloane, Jul 16 2005

Status

approved