A381673 -id:A381673 - cp's OEIS frontend

A089139 Decimal expansion of e^4 - 3e^3 + 2e^2 - e/6.

8, 6, 6, 6, 6, 0, 4, 4, 9, 0, 0, 3, 2, 6, 9, 5, 4, 3, 7, 2, 2, 5, 4, 7, 9, 2, 4, 8, 3, 7, 3, 6, 2, 9, 9, 2, 1, 8, 9, 4, 7, 7, 0, 1, 4, 8, 4, 3, 8, 6, 5, 3, 0, 1, 1, 7, 0, 2, 8, 8, 5, 6, 4, 3, 2, 1, 4, 9, 2, 5, 9, 5, 2, 7, 5, 9, 1, 3, 9, 2, 1, 5, 7, 3, 6, 8, 8, 3, 6, 8, 8, 2, 5, 6, 3, 9, 6, 8, 8, 7, 9, 6, 6, 2, 2

Offset: 1

Brian Dunfield (brian.dunfield(AT)sympatico.ca), Dec 05 2003

Expected number of picks from a uniform [0,1] needed to first exceed a sum of 4.

			8.66660449003269543722547924837362992189477014843865301170288564321492595275913921...

J. V. Uspensky, Introduction to Mathematical Probability, New York: McGraw-Hill, 1937.

Daniel Mondot, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Uniform Sum Distribution
Index entries for transcendental numbers

Cf. A001113, A090142, A090143, A090611, A379601, A381673, A382020, A381843, A382026, A090137, A090138, A089087.

RealDigits[ E^4 - 3E^3 + 2E^2 - E/6, 10, 111][[1]] (* Robert G. Wilson v, Dec 05 2003 *)

subst(x^4-3*x^3+2*x^2-x/6, x, exp(1)) \\ Charles R Greathouse IV, Dec 06 2016

Equals Sum_{k=0..n} (-1)^k * (n-k+1)^k * exp(n-k+1) / k! for n = 3 (Uspensky, 1937, p. 278).

Edited and extended by Robert G. Wilson v and Ray Chandler, Dec 07 2003

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.

Original entry on oeis.org

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

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

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

			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,
  ...

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.

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

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

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 *)

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

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).

Corrected and extended by Vladeta Jovovic, Dec 05 2003

A090611 Decimal expansion of (24e^5 - 96e^4 + 108e^3 - 32e^2 + e)/24.

Original entry on oeis.org

1, 0, 6, 6, 6, 6, 6, 2, 0, 6, 8, 6, 2, 2, 4, 1, 1, 8, 5, 8, 0, 1, 9, 0, 2, 7, 3, 7, 1, 9, 3, 2, 8, 4, 7, 0, 6, 8, 6, 0, 3, 1, 0, 2, 5, 8, 1, 0, 8, 4, 7, 5, 8, 3, 4, 3, 3, 2, 5, 7, 9, 3, 1, 9, 8, 1, 3, 3, 9, 6, 1, 0, 0, 4, 1, 2, 1, 6, 3, 4, 0, 7, 5, 3, 2, 8, 7, 8, 2, 0, 4, 3, 9, 2, 5, 0, 5, 0, 4, 3, 7, 9, 5, 8, 7

Offset: 2

Ray Chandler, Dec 07 2003

Expected number of picks from a uniform [0,1] needed to first exceed a sum of 5.

			10.66666206862241185801902737193284706860310258108475834332579319813396100412163407...

J. V. Uspensky, Introduction to Mathematical Probability, New York: McGraw-Hill, 1937.

Daniel Mondot, Table of n, a(n) for n = 2..10001
Eric Weisstein's World of Mathematics, Uniform Sum Distribution.
Index entries for transcendental numbers.

Cf. A001113, A090142, A090143, A089139, A379601, A381673, A382020, A381843, A382026, A090137, A090138, A089087.

RealDigits[E^5 - 4*E^4 + 9*E^3/2 - 4*E^2/3 + E/24, 10, 120][[1]] (* Amiram Eldar, Jun 20 2023 *)

exp(5)-4*exp(4)+9*exp(3)/2-4*exp(2)/3+exp(1)/24

Equals Sum_{k=0..n} (-1)^k * (n-k+1)^k * exp(n-k+1) / k! for n = 4 (Uspensky, 1937, p. 278).

Offset corrected by R. J. Mathar, Feb 05 2009

A379601 Decimal expansion of (120e^6 - 600e^5 + 960e^4 - 540e^3 + 80e^2 - e) / 120.

Original entry on oeis.org

1, 2, 6, 6, 6, 6, 6, 7, 1, 4, 1, 3, 7, 8, 1, 2, 1, 4, 0, 1, 3, 7, 1, 9, 3, 5, 7, 6, 2, 6, 8, 4, 9, 1, 1, 1, 9, 5, 6, 4, 7, 4, 3, 7, 0, 7, 7, 7, 4, 0, 1, 9, 6, 7, 5, 6, 7, 1, 0, 5, 3, 7, 5, 5, 6, 8, 2, 6, 0, 2, 8, 7, 6, 9, 4, 0, 6, 7, 8, 4, 2, 4, 8, 7, 0, 0, 5, 6, 0, 0, 9, 8, 0, 3, 5, 2, 2, 4, 0, 2, 0, 7, 8, 0, 7, 5, 9, 7, 6, 1, 6

Offset: 2

Daniel Mondot, Feb 27 2025

Expected number of picks from a uniform [0,1] needed to first exceed a sum of 6.

			12.6666671413781214013719357626849111...

J. V. Uspensky, Introduction to Mathematical Probability, New York: McGraw-Hill, 1937.

Daniel Mondot, Table of n, a(n) for n = 2..10001
Eric Weisstein's World of Mathematics, Uniform Sum Distribution.
Index entries for transcendental numbers.

Cf. A001113, A090142, A090143, A089139, A090611, A381673, A089087.

RealDigits[E^6 - 5*E^5 + 8*E^4 - 9*E^3/2 + 2*E^2/3 - E/120, 10, 120][[1]]

exp(6)-5*exp(5)+8*exp(4)-9*exp(3)/2+2*exp(2)/3-exp(1)/120

Equals Sum_{k=0..n} (-1)^k * (n-k+1)^k * exp(n-k+1) / k! for n = 5 (Uspensky, 1937, p. 278).

A381843 Decimal expansion of (40320e^9 - 322560e^8 + 987840e^7 - 1451520e^6 + 1050000e^5 - 344064e^4 + 40824e^3 - 1024e^2 + e) / 40320.

Original entry on oeis.org

1, 8, 6, 6, 6, 6, 6, 6, 6, 6, 5, 2, 7, 0, 3, 2, 1, 3, 4, 8, 9, 5, 5, 5, 2, 1, 7, 2, 2, 9, 4, 8, 5, 6, 9, 6, 1, 0, 0, 2, 7, 8, 4, 8, 3, 5, 6, 2, 1, 5, 5, 0, 7, 6, 9, 8, 4, 1, 6, 0, 8, 4, 6, 7, 9, 9, 2, 7, 1, 6, 2, 2, 2, 5, 3, 5, 9, 5, 2, 6, 2, 6, 5, 8, 1, 1, 3

Offset: 2

Daniel Mondot, Mar 12 2025

Expected number of picks from a uniform [0,1] distribution needed to first exceed a sum of 9.

			18.66666666527032134895552...

J. V. Uspensky, Introduction to Mathematical Probability, New York: McGraw-Hill, 1937.

Daniel Mondot, Table of n, a(n) for n = 2..10001

Cf. A001113, A090142, A090143, A089139, A090611, A379601, A381673, A382020, A382026, A089087.

RealDigits[E^9 - 8*E^8 + 49*E^7/2 - 36*E^6 + 625*E^5/24 - 128*E^4/15 + 81*E^3/80 - 8*E^2/315 + E/40320, 10, 120][[1]]

exp(9)-8*exp(8)+49*exp(7)/2-36*exp(6)+625*exp(5)/24-128*exp(4)/15+81*exp(3)/80-8*exp(2)/315+exp(1)/40320

Equals Sum_{k=0..n} (-1)^k * (n-k+1)^k * exp(n-k+1) / k! for n = 8 (Uspensky, 1937, p. 278).

A382020 Decimal expansion of (5040e^8 - 35280e^7 + 90720e^6 - 105000e^5 + 53760e^4 - 10206e^3 + 448*e^2 - e) / 5040.

Original entry on oeis.org

1, 6, 6, 6, 6, 6, 6, 6, 6, 7, 0, 4, 2, 6, 8, 8, 7, 8, 2, 3, 6, 6, 2, 3, 4, 7, 0, 0, 4, 3, 3, 2, 5, 8, 0, 4, 4, 9, 3, 6, 4, 9, 5, 7, 7, 5, 8, 9, 7, 0, 2, 0, 7, 0, 7, 8, 7, 1, 2, 8, 4, 1, 5, 7, 6, 3, 7, 6, 1, 8, 5, 7, 5, 9, 4, 9, 7, 2, 1, 4, 6, 2, 7, 6, 4, 6, 6, 0

Offset: 2

Daniel Mondot, Mar 12 2025

Expected number of picks from a uniform [0,1] distribution needed to first exceed a sum of 8.

			16.6666666704268878236623470...

J. V. Uspensky, Introduction to Mathematical Probability, New York: McGraw-Hill, 1937.

Daniel Mondot, Table of n, a(n) for n = 2..10001

Cf. A001113, A090142, A090143, A089139, A090611, A379601, A381673, A381843, A382026, A089087.

RealDigits[E^8 - 7*E^7 + 18*E^6 - 125*E^5/6 + 32*E^4/3 - 81*E^3/40 + 4*E^2/45 - E/5040, 10, 120][[1]]

exp(8)-7*exp(7)+18*exp(6)-125*exp(5)/6+32*exp(4)/3-81*exp(3)/40+4*exp(2)/45-exp(1)/5040

Equals Sum_{k=0..n} (-1)^k * (n-k+1)^k * exp(n-k+1) / k! for n = 7 (Uspensky, 1937, p. 278).

A382026 Decimal expansion of (362880e^10 - 3265920e^9 + 11612160e^8 - 20744640e^7 + 19595520e^6 - 9450000e^5 + 2064384e^4 - 157464e^3 + 2304*e^2 - e) / 362880.

Original entry on oeis.org

2, 0, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4, 7, 6, 3, 1, 8, 8, 0, 0, 6, 1, 4, 1, 6, 3, 0, 9, 1, 0, 5, 9, 7, 6, 6, 4, 6, 8, 6, 5, 6, 8, 6, 0, 8, 2, 1, 5, 4, 4, 7, 4, 2, 3, 8, 4, 1, 9, 2, 0, 9, 0, 6, 0, 0, 0, 7, 3, 8, 5, 3, 6, 8, 8, 3, 6, 1, 5, 8, 9, 8, 2, 5, 8, 2, 3, 4, 5

Offset: 2

Daniel Mondot, Mar 12 2025

Expected number of picks from a uniform [0,1] distribution needed to first exceed a sum of 10.

			20.666666666476318800614163...

J. V. Uspensky, Introduction to Mathematical Probability, New York: McGraw-Hill, 1937.

Daniel Mondot, Table of n, a(n) for n = 2..10001

Cf. A001113, A090142, A090143, A089139, A090611, A379601, A381673, A382020, A381843, A089087.

RealDigits[E^10 - 9*E^9 + 32*E^8 - 343*E^7/6 + 54*E^6 - 625*E^5/24 + 256*E^4/45 - 243*E^3/560 + 2*E^2/315 - E/362880, 10, 120][[1]]

exp(10)-9*exp(9)+32*exp(8)-343*exp(7)/6+54*exp(6)-625*exp(5)/24+256*exp(4)/45-243*exp(3)/560+2*exp(2)/315-exp(1)/362880

Equals Sum_{k=0..n} (-1)^k * (n-k+1)^k * exp(n-k+1) / k! for n = 9 (Uspensky, 1937, p. 278).

cp's OEIS Frontend

A089139 Decimal expansion of e^4 - 3*e^3 + 2*e^2 - e/6.

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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.

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A090611 Decimal expansion of (24*e^5 - 96*e^4 + 108*e^3 - 32*e^2 + e)/24.

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A379601 Decimal expansion of (120e^6 - 600e^5 + 960e^4 - 540e^3 + 80e^2 - e) / 120.

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A381843 Decimal expansion of (40320*e^9 - 322560*e^8 + 987840*e^7 - 1451520*e^6 + 1050000*e^5 - 344064*e^4 + 40824*e^3 - 1024*e^2 + e) / 40320.

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A382020 Decimal expansion of (5040*e^8 - 35280*e^7 + 90720*e^6 - 105000*e^5 + 53760*e^4 - 10206*e^3 + 448*e^2 - e) / 5040.

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A089139 Decimal expansion of e^4 - 3e^3 + 2e^2 - e/6.

A090611 Decimal expansion of (24e^5 - 96e^4 + 108e^3 - 32e^2 + e)/24.

A381843 Decimal expansion of (40320e^9 - 322560e^8 + 987840e^7 - 1451520e^6 + 1050000e^5 - 344064e^4 + 40824e^3 - 1024e^2 + e) / 40320.

A382020 Decimal expansion of (5040e^8 - 35280e^7 + 90720e^6 - 105000e^5 + 53760e^4 - 10206e^3 + 448*e^2 - e) / 5040.

A382026 Decimal expansion of (362880e^10 - 3265920e^9 + 11612160e^8 - 20744640e^7 + 19595520e^6 - 9450000e^5 + 2064384e^4 - 157464e^3 + 2304*e^2 - e) / 362880.