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

Offset: 0

Author

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

Keywords

• easy
• sign
• tabl

Comments

Expected number of uniform random choices of X from interval[0,1] so that their sum exceeds ...

1 is e/0!,

2 is (e^2-e)/1!,

3 is (2e^3-4e^2+e)/2!.

Examples

			Triangle begins:
       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, -9450000, 2064384, -157464, 2304, -1,
  ...
		

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

• Daniel Mondot, Table of n, a(n) for n = 0..5049
• Eric Weisstein's World of Mathematics, Uniform Sum Distribution.

Crossrefs

Cf. A001113, A090142, A090143, A089139, A090611, A379601, A381673, A382020, A381843, A382026, A090137, A090138.

Programs

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 *)

Sage

def A089087_row(n):
    R. = 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

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).

Extensions

Corrected and extended by Vladeta Jovovic, Dec 05 2003