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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.

%I A089087 #38 Mar 23 2025 13:47:46
%S A089087 1,1,-1,2,-4,1,6,-18,12,-1,24,-96,108,-32,1,120,-600,960,-540,80,-1,
%T A089087 720,-4320,9000,-7680,2430,-192,1,5040,-35280,90720,-105000,53760,
%U A089087 -10206,448,-1,40320,-322560,987840,-1451520,1050000,-344064,40824,-1024,1,362880,-3265920,11612160,-20744640,19595520
%N 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.
%C A089087 Expected number of uniform random choices of X from interval[0,1] so that their sum exceeds ...
%C A089087 1 is e/0!,
%C A089087 2 is (e^2-e)/1!,
%C A089087 3 is (2e^3-4e^2+e)/2!.
%D A089087 J. Derbyshire, "Prime Obsession: Bernhard Riemann and the Greatest Unsolved...", Henry Press, 2003, footnote on page 366.
%D A089087 J. V. Uspenski, "Introduction to Mathematical Probability", McGraw Hill, 1937, p. 278.
%H A089087 Daniel Mondot, <a href="/A089087/b089087.txt">Table of n, a(n) for n = 0..5049</a>
%H A089087 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UniformSumDistribution.html">Uniform Sum Distribution</a>.
%F A089087 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 A089087 E.g.f.: 1/(exp(y*x)-x).
%e A089087 Triangle begins:
%e A089087        1,
%e A089087        1,       -1,
%e A089087        2,       -4,       1,
%e A089087        6,      -18,      12,         -1,
%e A089087       24,      -96,     108,        -32,        1,
%e A089087      120,     -600,     960,       -540,       80,       -1,
%e A089087      720,    -4320,    9000,      -7680,     2430,     -192,       1,
%e A089087     5040,   -35280,   90720,    -105000,    53760,   -10206,     448,      -1,
%e A089087    40320,  -322560,   987840,  -1451520,  1050000,  -344064,   40824,   -1024,    1,
%e A089087   362880, -3265920, 11612160, -20744640, 19595520, -9450000, 2064384, -157464, 2304, -1,
%e A089087   ...
%t A089087 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 *)
%t A089087 Table[n!*CoefficientList[f[n], E] // Reverse // Most, {n, 0, 9}] // Flatten (* _Jean-François Alcover_, Nov 05 2013 *)
%o A089087 (Sage)
%o A089087 def A089087_row(n):
%o A089087     R.<x> = ZZ[]
%o A089087     P = add((n-k+1)^k*x^(n-k+1)*factorial(n)/factorial(k) for k in (0..n))
%o A089087     return [(-1)^i*P[n-i+1] for i in (0..n)]
%o A089087 for n in (0..5): print(A089087_row(n))  # _Peter Luschny_, May 03 2013
%Y A089087 Cf. A001113, A090142, A090143, A089139, A090611, A379601, A381673, A382020, A381843, A382026, A090137, A090138.
%K A089087 easy,sign,tabl
%O A089087 0,4
%A A089087 Brian Dunfield (brian.dunfield(AT)sympatico.ca), Dec 04 2003
%E A089087 Corrected and extended by _Vladeta Jovovic_, Dec 05 2003