A055775 -id:A055775 - cp's OEIS frontend

A210116 Floor of the expected value of number of trials until exactly five cells are empty in a random distribution of n balls in n cells.

7776, 311, 51, 16, 7, 4, 3, 3, 2, 3, 3, 4, 5, 8, 11, 16, 25, 40, 66, 110, 187, 325, 574, 1032, 1885, 3492, 6557, 12467, 23988, 46667, 91731, 182078, 364734, 736972, 1501318, 3082136, 6374007, 13273719, 27825438, 58697777, 124566798

Offset: 6

Washington Bomfim, Mar 18 2012

nonn

Also floor of the expected value of number of trials until we have n-5 distinct symbols in a random sequence on n symbols of length n. A055775 corresponds to zero cells empty.

			For n=6, there are 6^6 = 46656 sequences on 6 symbols of length 6. Only 6 sequences has a unique symbol, so a(6) = floor(46656/6) = 7776.

W. Feller, An Introduction to Probability Theory and its Applications, 2nd ed, Wiley, New York, 1965, (2.4) p. 92. (Occupancy problems)

W. Bomfim, Table of n, a(n) for n = 6..100

Cf. A055775, A209899, A209900, A210112, A210113, A210114, A210115.

With m = 5, a(n) = floor(n^n/(binomial(n,m)*_Sum{v=0..n-m-1}((-1)^v*binomial(n-m,v)*(n-m-v)^n)))

A121563 Numerator of Sum_{i=1..n} i!/(i^i).

Original entry on oeis.org

0, 1, 3, 31, 523, 333787, 3029083, 357534985867, 1466339436008107, 86629054728081110243, 54153614078975016066611, 1404709830784309760237846406097211, 1404749977001540310126014452972211

Offset: 0

Jonathan Vos Post, Aug 07 2006

			A121563/A121564 = 0, 1, 3/2, 31/18, 523/288, 333787/180000, ... .

Denominators are in A121564.

Cf. A000142, A000312, A055775, A094082.

Numerator[Table[Sum[i!/(i^i),{i,n}],{n,12}]] (* James C. McMahon, Oct 19 2024 *)

Numerator of Sum_{i=1..n} A000142(i)/A000312(i).

Limit_{n -> oo} A121563(n)/A121564(n) = A094082.

A121564 Denominator of Sum_{i=1..n} i!/(i^i).

Original entry on oeis.org

1, 1, 2, 18, 288, 180000, 1620000, 190591380000, 780662292480000, 46097327708651520000, 28810829817907200000000, 747278726094210559615027200000000, 747278726094210559615027200000000, 17410163370762081276553474535120146483200000000

Offset: 0

Jonathan Vos Post, Aug 07 2006

			A121563/A121564 = 0, 1, 3/2, 31/18, 523/288, 333787/180000, ... .

Numerators are in A121563.

Cf. A000142, A000312, A055775, A094082.

Denominator[Table[Sum[i!/(i^i),{i,n}],{n,12}]] (* James C. McMahon, Oct 19 2024 *)

Denominator of Sum_{i=1..n} A000142(i)/A000312(i).

Limit_{n -> oo} A121563(n)/A121564(n) = A094082.

A208847 A056915(n) mod 5228905 mod 17.

Original entry on oeis.org

3, 4, 13, 15, 8, 14, 9, 5, 0, 11, 16, 10, 2, 12, 7, 1, 6, 16, 3, 10, 5, 8, 7, 16, 6, 11, 13, 6, 10, 6, 11, 16, 9, 1, 1, 15, 5, 1, 14, 7, 15, 2, 14, 9, 2, 6, 14, 3, 3, 14, 12, 6, 2, 4, 10, 16, 6, 10, 9, 3, 3, 1, 7, 9, 11, 5

Offset: 1

Washington Bomfim, Mar 02 2012

nonn

A056915(n) mod 5228905 mod 17 is a bijection from the set of the first 17 terms of A056915 to {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}.
From an algorithm based on strong pseudoprimes to bases 2,3 and 5, and a table T with the first 17 terms of A056915, we can test if n is prime, odd n, 1 < n < 42550716781. When n is a prime, we check if n belongs to T. A fast way to do that is to compute i = n mod 5228905 mod 17 and compare n with T[i]. If n is not equal to T[i], n is prime.
Terms computed using table by Charles R Greathouse IV. See A056915.

Washington Bomfim, A method to find bijections from a set of n integers to {0,1, ... ,n-1}

Cf. A056915, A055775, A208846.

A215460 a(n) = floor(n!^2 / n^n).

Original entry on oeis.org

1, 1, 1, 2, 4, 11, 30, 96, 339, 1316, 5584, 25733, 128025, 683949, 3905083, 23731187, 152934464, 1041782238, 7479469995, 56448098958, 446768591341, 3700276748921, 32007269639380, 288630046441757, 2708888570942365, 26419890078249485, 267389254029561667, 2804508541393392135, 30446382653772707171, 341729529206733994569

Offset: 1

Alex Ratushnyak, Aug 11 2012

nonn

			a(4) = floor((4!)^2 / (4^4)) = floor(24^2 / 256) = floor(2.25) = 2.

Paolo Xausa, Table of n, a(n) for n = 1..500

Cf. A064546, A063709, A055775.

A215460[n_] := Quotient[n!^2, n^n];
Array[A215460, 30] (* Paolo Xausa, Aug 05 2025 *)

f = 1
for n in range(1,33):
    print(f*f // n**n)
    f *= n+1

More terms from Paolo Xausa, Aug 05 2025

A235496 a(n) = round(n^n/n!) where round(1/2)=1.

Original entry on oeis.org

1, 1, 2, 5, 11, 26, 65, 163, 416, 1068, 2756, 7148, 18614, 48639, 127463, 334865, 881658, 2325751, 6145597, 16263866, 43099804, 114356611, 303761260, 807692035, 2149632061, 5726042115, 15264691107, 40722913454, 108713644516, 290404350963, 776207020880

Offset: 0

Vincenzo Librandi, Jan 15 2014

We have two versions of this sequence, this and A240571, because there is no universal agreement on how to round a number like 9/2. - N. J. A. Sloane, Dec 13 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000142, A000312, A055775, A073225.

See A240571 for another version.

[Round(n^n/Factorial(n)): n in [1..40]];

a:= n-> round(n^n/n!):
seq(a(n), n=0..32);  # Alois P. Heinz, Dec 13 2015

Mathematica
```
Table[Floor[n^n/n! + 1/2], {n, 40}]
```

s=[]; for(n=1, 30, s=concat(s, round(n^n/n!))); s \\ Colin Barker, Jan 19 2014

a(n) = Round(A000312(n)/A000142(n)).

Name clarified by Sean A. Irvine, Jan 12 2025

A114853 a(n) = floor(n^n/n!!).

Original entry on oeis.org

1, 2, 9, 32, 208, 972, 7843, 43690, 409968, 2604166, 27447010, 193491763, 2241278030, 17224712961, 216027868615, 1787142709274, 24006211998207, 211773735868781, 3021737893128258, 28218694885361552, 424936725846414486

Offset: 1

Jonathan Vos Post, Feb 20 2006

This is to double factorial A006882 as A055775 "Floor(n^n/n!)" is to factorial. This sequence is a weak first approximation of a double factorial analog to Stirling's approximation to factorial. Note that a(n) is exact for n = 1, 2, 3, 4, 6.

			a(10) = floor((10^10)/3840) = floor(2604166.67) = 2604166.

Cf. A000312, A006882, A055775.

A114853 := proc(n)
    n^n/doublefactorial(n) ;
    floor(%) ;
end proc:
seq(A114853(n),n=1..25) ; # R. J. Mathar, Jun 23 2014

Table[Floor[n^n/n!!],{n,30}] (* Harvey P. Dale, Jul 29 2023 *)

a(n) = floor(n^n/n!!). a(n) = floor(A000312(n)/A006882(n)).

A114854 a(n) = floor(n^(n/2)/n!!).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 8, 10, 20, 26, 51, 64, 128, 163, 326, 416, 834, 1067, 2148, 2755, 5559, 7147, 14449, 18613, 37696, 48638, 98650, 127463, 258857, 334864, 680822, 881657, 1794294, 2325750, 4737361

Offset: 1

Jonathan Vos Post, Feb 20 2006

This sequence is a second approximation of a double factorial analog to Stirling's approximation to factorial. Note that a(n) is exact for n = 1, 2, 4.

			a(10) = floor((10^5)/3840) = floor(26.0416667) = 26.
a(11) = floor((11^5.5)/10395) = floor(51.3848715) = 51.

Cf. A000312, A006882, A055775.

A114854 := proc(n)
    n^(n/2)/doublefactorial(n) ;
    floor(%) ;
end proc:
seq(A114854(n),n=1..35) ; # R. J. Mathar, Jun 23 2014

Table[Floor[n^(n/2)/n!!],{n,40}] (* Harvey P. Dale, Apr 04 2019 *)

a(n) = floor(n^(n/2)/n!!). a(n) = floor(sqrt(A000312(n))/A006882(n)).

A114863 a(n) = floor(n^(n/3)/n!!!).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 4, 7, 7, 10, 18, 18, 26, 45, 44, 64, 113, 112, 163, 287, 285, 416, 733, 731, 1067, 1885, 1885, 2755, 4873, 4881, 7147, 12659, 12697, 18613, 33007, 33143, 48638, 86337, 86777, 127463, 226454, 227795, 334864, 595382, 599342, 881657

Offset: 1

Jonathan Vos Post, Feb 20 2006

This sequence is an approximation of a triple factorial analog to Stirling's approximation to factorial. Note that a(n) is exact for n = 1, 3, 6.

			a(9) = floor((9^3)/162) = floor(4.5) = 4.
a(10) = floor((10^3.33333)/280) = floor(7.69381906) = 7.
a(27) = floor((27^9)/7142567040) = floor(1067.62701) = 1067.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000312, A006882, A055775, A007661.

fac[n_, m_] := Block[{t = n, f = Max[1, n]}, While[t > m, t -= m; f *= t]; f]; a[n_] := Floor[n^(n/3)/fac[n, 3]]; Array[a, 40] (* Giovanni Resta, Jun 15 2016 *)
Table[Floor[n^(n/3)/Times@@Range[n,1,-3]],{n,50}] (* Harvey P. Dale, Aug 06 2021 *)

a(n) = floor(n^(n/3)/n!!!). a(n) = floor((A000312(n)^(1/3))/A007661(n)).

More terms from Giovanni Resta, Jun 15 2016

A114868 a(n) = floor(n^(n/4)/n!!!!).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 2, 3, 2, 3, 4, 7, 6, 7, 10, 17, 14, 18, 26, 41, 36, 44, 64, 104, 91, 112, 163

Offset: 1

Jonathan Vos Post, Feb 20 2006

This sequence is an approximation to a quadruple factorial analog of Stirling's approximation to the factorial function. Note that a(n) is exact for n = 1, 4, 8.

			a(8) = floor((8^2)/8!!!!) = floor((8^2)/32) = floor(2) = 2.
a(9) = floor((9^2.25)/9!!!!) = floor((9^2.25)/45) = floor(3.11769145) = 3.
a(16) = floor((16^4)/16!!!!) = floor((16^4)/6144) = floor(10.6666667) = 10.
a(20) = floor((20^5)/20!!!!) = floor((20^5)/122880) = floor(26.0416667) = 26.

Cf. A000312, A006882, A055775, A007662.

a(n) = floor(n^(n/4)/n!!!). a(n) = floor((A000312(n)^(1/4))/A007662(n)).

cp's OEIS Frontend

A210116 Floor of the expected value of number of trials until exactly five cells are empty in a random distribution of n balls in n cells.

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A121563 Numerator of Sum_{i=1..n} i!/(i^i).

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A121564 Denominator of Sum_{i=1..n} i!/(i^i).

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A208847 A056915(n) mod 5228905 mod 17.

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A215460 a(n) = floor(n!^2 / n^n).

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A235496 a(n) = round(n^n/n!) where round(1/2)=1.

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A114853 a(n) = floor(n^n/n!!).

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Maple

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A114854 a(n) = floor(n^(n/2)/n!!).

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