A089087 - OEIS

A089087

Triangular array of coefficients multiplied by n! of polynomials in e. These give the expected number of trials needed for the sum of uniform random variables from the interval [0,1] to exceed n+1.

1, 1, -1, 2, -4, 1, 6, -18, 12, -1, 24, -96, 108, -32, 1, 120, -600, 960, -540, 80, -1, 720, -4320, 9000, -7680, 2430, -192, 1, 5040, -35280, 90720, -105000, 53760, -10206, 448, -1, 40320, -322560, 987840, -1451520, 1050000, -344064, 40824, -1024, 1, 362880, -3265920, 11612160, -20744640, 19595520

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,4

REFERENCES

J. Derbyshire, "Prime Obsession: Bernhard Riemann and the Greatest Unsolved...", Henry Press, 2003, footnote on page 366.

J. V. Uspenski, "Introduction to Mathematical Probability", McGraw Hill, 1937, p. 278.

LINKS

Table of n, a(n) for n=0..49.

Eric Weisstein's World of Mathematics, Uniform Sum Distribution.

FORMULA

T(n,k) = (-1)^k*n!*(n+1-k)^k/k!; k-th coefficient of n-th row for n >= 0 and k >= 0.

E.g.f.: 1/(exp(y*x)-x).

EXAMPLE

Expected number of uniform random choices of X from interval[0,1] so that their sum exceeds 1 is e/0!. So that sum exceeds 2: (e^2-e)/1!. So that sum exceeds 3: (2e^3-4e^2+e)/2!.

Triangle begins:

1, -1,

2, -4, 1,

6, -18, 12, -1,

24, -96, 108, -32, 1,

...

MATHEMATICA

f[n_] := Sum[(-1)^k*(n-k+1)^k*E^(n-k+1)/k!, {k, 0, n}]; (* f(0)=A001113=e, f(1)=A090142, f(2)=A090143, f(3)=A089139, f(4)=A090611 *) Table[n!*CoefficientList[f[n], E] // Reverse // Most, {n, 0, 9}] // Flatten (* Jean-François Alcover, Nov 05 2013 *)

PROG

(Sage)

def A089087_row(n):

R.<x> = ZZ[]

P = add((n-k+1)^k*x^(n-k+1)*factorial(n)/factorial(k) for k in (0..n))

return [(-1)^i*P[n-i+1] for i in (0..n)]

for n in (0..5): print(A089087_row(n)) # Peter Luschny, May 03 2013

CROSSREFS

Cf. A001113, A090142, A090143, A089139, A090611, A090137, A090138.

Sequence in context: A204024 A021009 A137478 * A142146 A143350 A302541

Adjacent sequences: A089084 A089085 A089086 * A089088 A089089 A089090

KEYWORD

easy,sign,tabl

AUTHOR

Brian Dunfield (brian.dunfield(AT)sympatico.ca), Dec 04 2003

EXTENSIONS

Corrected and extended by Vladeta Jovovic, Dec 05 2003

STATUS

approved