A019576 -id:A019576 - cp's OEIS frontend

A019575 Place n distinguishable balls in n boxes (in n^n ways); let T(n,k) = number of ways that the maximum in any box is k, for 1 <= k <= n; sequence gives triangle of numbers T(n,k).

1, 2, 2, 6, 18, 3, 24, 180, 48, 4, 120, 2100, 800, 100, 5, 720, 28800, 14700, 2250, 180, 6, 5040, 458640, 301350, 52920, 5292, 294, 7, 40320, 8361360, 6867840, 1342600, 153664, 10976, 448, 8, 362880, 172141200, 172872000, 36991080, 4644864, 387072, 20736, 648, 9

Offset: 1

Lee Corbin (lcorbin(AT)tsoft.com)

T(n,k) is the number of endofunctions on [n] such that the maximal cardinality of the nonempty preimages equals k. - Alois P. Heinz, Jul 31 2014

			Triangle begins:
       1;
       2,         2;
       6,        18,         3;
      24,       180,        48,        4;
     120,      2100,       800,      100,       5;
     720,     28800,     14700,     2250,     180,      6;
    5040,    458640,    301350,    52920,    5292,    294,     7;
   40320,   8361360,   6867840,  1342600,  153664,  10976,   448,   8;
  362880, 172141200, 172872000, 36991080, 4644864, 387072, 20736, 648, 9;
  ...

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A019576. See A180281 for the case when the balls are indistinguishable.
Rows sums give A000312.
Cf. A245687.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-j, i-1, k)/j!, j=0..min(k, n))))
    end:
T:= (n, k)-> n!* (b(n$2, k) -b(n$2, k-1)):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Jul 29 2014

f[0, , b] := Boole[b == 0]; f[n_, k_, b_] := f[n, k, b] = Sum[ Binomial[b, i]*f[n - 1, k, b - i], {i, 0, Min[k, b]}]; t[n_, k_] := f[n, k, n] - f[n, k - 1, n]; Flatten[ Table[ t[n, k], {n, 1, 9}, {k, 1, n}]] (* Jean-François Alcover, Mar 09 2012, after Robert Gerbicz *)

/*setup memoization table for args <= M. Could be done dynamically inside f() */
M=10;F=vector(M,i,vector(M,i,vector(M)));
f(n,k,b)={ (!n||!b||!k) & return(!b); F[n][k][b] & return(F[n][k][b]);
F[n][k][b]=sum(i=0,min(k,b),binomial(b,i)*f(n-1,k,b-i)) }
T(n,k)=f(n,k,n)-f(n,k-1,n)
for(n=1,9,print(vector(n,k,T(n,k))))
\\ M. F. Hasler, Aug 19 2010; Based on Robert Gerbicz's code I suggest the following (very naively) memoized version of "f"

A019575(x, z) = Sum ( A049009(p)) where x = A036042(p), z = A049085(p) - Alford Arnold.
From Robert Gerbicz, Aug 19 2010: (Start)
Let f(n,k,b) = number of ways to place b balls to n boxes, where the max in any box is not larger than k. Then T(n,k) = f(n,k,n) - f(n,k-1,n). We have:
f(n, k, b) = if(n=0, if(b=0, 1, 0), Sum_{i=0..min(k, b)} binomial(b, i)*f(n-1, k, b-i)).
T(n,k) = f(n,k,n) - f(n,k-1,n). (End)

Edited by N. J. A. Sloane, Sep 06 2010

A019577 Place n distinguishable balls in n boxes (in n^n ways); let f(n,k) = number of ways that max in any box is k, for 1<=k<=n; sequence gives f(n,2)/n.

Original entry on oeis.org

0, 1, 6, 45, 420, 4800, 65520, 1045170, 19126800, 395448480, 9120988800, 232248416400, 6471820555200, 195912193276800, 6402199349145600, 224636583525354000, 8423131243022496000, 336138596955120960000, 14224375944427993344000, 636224790017466730080000

Offset: 1

Lee Corbin (lcorbin(AT)tsoft.com), N. J. A. Sloane

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A019576.

Sum n! (n-1)! / ( 2^d (n-2d)! d! d! ), d=1..[ n/2 ].

A019579 Place n distinguishable balls in n boxes (in n^n ways); let f(n,k) = number of ways that max in any box is k, for 1 <= k <= n; sequence gives f(n,n-2)/n.

Original entry on oeis.org

2, 45, 160, 375, 756, 1372, 2304, 3645, 5500, 7986, 11232, 15379, 20580, 27000, 34816, 44217, 55404, 68590, 84000, 101871, 122452, 146004, 172800, 203125, 237276, 275562, 318304, 365835, 418500, 476656, 540672, 610929, 687820, 771750, 863136, 962407

Offset: 3

Lee Corbin (lcorbin(AT)tsoft.com), N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A019576.

CoefficientList[Series[-(9 x^6 - 35 x^5 + 41 x^4 + 5 x^3 - 45 x^2 + 35 x + 2)/(x - 1)^5, {x, 0, 50}], x] (* Vincenzo Librandi, Oct 16 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{2,45,160,375,756,1372,2304},40] (* Harvey P. Dale, Dec 18 2020 *)

a(n) = n*(n-1)^3/2, n >= 5.
G.f.: -x^3*(9*x^6 - 35*x^5 + 41*x^4 + 5*x^3 - 45*x^2 + 35*x + 2) / (x-1)^5. [Colin Barker, Jan 11 2013]
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Wesley Ivan Hurt, Dec 27 2021

More terms from Vincenzo Librandi, Oct 16 2013

A019581 Place n distinguishable balls in n boxes (in n^n ways); let f(n,k) = number of ways that max in any box is k, for 1<=k<=n; sequence gives f(n,2).

Original entry on oeis.org

0, 2, 18, 180, 2100, 28800, 458640, 8361360, 172141200, 3954484800, 100330876800, 2786980996800, 84133667217600, 2742770705875200, 96032990237184000, 3594185336405664000, 143193231131382432000, 6050494745192177280000, 270263142944131873536000

Offset: 1

Lee Corbin (lcorbin(AT)tsoft.com), N. J. A. Sloane

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A019576.

Column k=2 of A019575. - Alois P. Heinz, Jul 29 2014

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-j, i-1)/j!, j=0..min(2, n))))
    end:
a:= n-> n! *(b(n$2) -1):
seq(a(n), n=1..30);  # Alois P. Heinz, Jul 29 2014

a[n_] := n! (Hypergeometric2F1[1/2 - n/2, -n/2, 1, 2] - 1); Array[a, 30] (* Jean-François Alcover, Feb 18 2016 *)

a(n) = sum(d=1, n\2, n!^2 / (2^d * (n-2*d)! * d!^2)); \\ Michel Marcus, Aug 13 2013

a(n) = sum(d=1..floor(n/2), n!^2 / ( 2^d * (n-2*d)! * d! * d! ) ).

A019578 Place n distinguishable balls in n boxes (in n^n ways); let f(n,k) = number of ways that max in any box is k, for 1<=k<=n; sequence gives f(n,3)/n.

Original entry on oeis.org

1, 12, 160, 2450, 43050, 858480, 19208000, 477237600, 13049190000, 389642022000, 12620614406400, 440871877446000, 16525943429595600, 661779275661504000, 28199857953915648000, 1274233240573028736000, 60863881753459328544000

Offset: 3

Lee Corbin (lcorbin(AT)tsoft.com)

nonn

Cf. A019576.

More terms from Sean A. Irvine, Mar 27 2019

A019580 Place n distinguishable balls in n boxes (in n^n ways); let f(n,k) = number of ways that max in any box is k, for 1<=k<=n; sequence gives f(n,4)/n.

Original entry on oeis.org

1, 20, 375, 7560, 167825, 4110120, 110602800, 3252075750, 103881924300, 3585467886000, 133052005386300, 5284678706307000, 223761183682286250, 10063370411081988000, 479152851159102120000, 24082014064135799772000

Offset: 4

Lee Corbin (lcorbin(AT)tsoft.com)

nonn

Cf. A019576.

More terms from Sean A. Irvine, Mar 27 2019

cp's OEIS Frontend

A019575 Place n distinguishable balls in n boxes (in n^n ways); let T(n,k) = number of ways that the maximum in any box is k, for 1 <= k <= n; sequence gives triangle of numbers T(n,k).

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A019577 Place n distinguishable balls in n boxes (in n^n ways); let f(n,k) = number of ways that max in any box is k, for 1<=k<=n; sequence gives f(n,2)/n.

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A019579 Place n distinguishable balls in n boxes (in n^n ways); let f(n,k) = number of ways that max in any box is k, for 1 <= k <= n; sequence gives f(n,n-2)/n.

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A019581 Place n distinguishable balls in n boxes (in n^n ways); let f(n,k) = number of ways that max in any box is k, for 1<=k<=n; sequence gives f(n,2).

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Maple

Mathematica

PARI

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A019578 Place n distinguishable balls in n boxes (in n^n ways); let f(n,k) = number of ways that max in any box is k, for 1<=k<=n; sequence gives f(n,3)/n.

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Crossrefs

Extensions

A019580 Place n distinguishable balls in n boxes (in n^n ways); let f(n,k) = number of ways that max in any box is k, for 1<=k<=n; sequence gives f(n,4)/n.

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