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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.
2

%I #27 Mar 04 2023 11:52:17

%S 1,1,-1,2,-4,1,6,-18,12,-1,24,-96,108,-32,1,120,-600,960,-540,80,-1,

%T 720,-4320,9000,-7680,2430,-192,1,5040,-35280,90720,-105000,53760,

%U -10206,448,-1,40320,-322560,987840,-1451520,1050000,-344064,40824,-1024,1,362880,-3265920,11612160,-20744640,19595520

%N 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.

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

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UniformSumDistribution.html">Uniform Sum Distribution</a>.

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

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

%e 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!.

%e Triangle begins:

%e 1,

%e 1, -1,

%e 2, -4, 1,

%e 6, -18, 12, -1,

%e 24, -96, 108, -32, 1,

%e ...

%t 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 *)

%o (Sage)

%o def A089087_row(n):

%o R.<x> = ZZ[]

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

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

%o for n in (0..5): print(A089087_row(n)) # _Peter Luschny_, May 03 2013

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

%K easy,sign,tabl

%O 0,4

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

%E Corrected and extended by _Vladeta Jovovic_, Dec 05 2003