A187783 -id:A187783 - cp's OEIS frontend

A248814 a(n) = (6n)!/(6!^n).

1, 1, 924, 17153136, 2308743493056, 1370874167589326400, 2670177736637149247308800, 14007180988362844601443040716800, 171889289584866507880743491472699801600, 4439413043841128802009762476941510771390464000

Offset: 0

Tilman Piesk, Oct 29 2014

Column 6 of A187783.

Number of permutations of a multiset that contains n different elements, each occurring 6 times.

			a(3) = (6*3)!/(6!^3) = 17153136 is the number of permutations of a multiset that contains 3 different elements 6 times, e.g., {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}.

Table of n, a(n) for n = 0..54

Cf. A187783.

[Factorial(6*n)/(720^n) : n in [0..10]]; // Wesley Ivan Hurt, Nov 01 2014

A248814:=n->(6*n)!/(720^n): seq(A248814(n), n=0..10); Wesley Ivan Hurt, Nov 01 2014

Table[(6 n)!/(720^n), {n, 0, 10}] (* Wesley Ivan Hurt, Nov 01 2014 *)

a(n) = (6n)!/(6!^n).

A172603 a(n) = (7n)!/(7!^n).

Original entry on oeis.org

1, 1, 3432, 399072960, 472518347558400, 3177459078523411968000, 85722533226982363751829504000, 7363615666157189603982585462030336000, 1707750599894443404262670865631874246246400000

Offset: 0

R. H. Hardin, Feb 06 2010

From Tilman Piesk, Oct 30 2014: (Start)
Column 7 of A187783.
Number of permutations of a multiset that contains n different elements 7 times.
Or in other words (the former title of this sequence):
Number of 7*n X n 0..1 arrays with row sums 1 and column sums 7.
(End)

			a(3) = (7*3)!/(7!^3) = 399072960 is the number of permutations of a multiset that contains 3 different elements 7 times, e.g., {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3}.

Tilman Piesk, Table of n, a(n) for n = 0..54 (first 14 terms from R. H. Hardin)

[Factorial(7*n)/(5040^n):  n in [0..20]]; // Vincenzo Librandi, Nov 01 2014

A172603:=n->(7*n)!/(5040^n): seq(A172603(n), n=0..10); # Wesley Ivan Hurt, Nov 01 2014

Table[(7 n)! / (5040^n), {n, 0, 10}] (* Vincenzo Librandi, Nov 01 2014 *)

a(n) = (7n)!/(7!^n).

Name changed by Tilman Piesk, Oct 30 2014

A172609 a(n) = (8n)!/(8!^n).

Original entry on oeis.org

1, 1, 12870, 9465511770, 99561092450391000, 7656714453153197981835000, 2889253496242619386328267523990000, 4104167472585675600759440022842715359250000, 18165723931630806756964027928179555634194028454000000

Offset: 0

R. H. Hardin, Feb 06 2010

From Tilman Piesk, Oct 30 2014: (Start)
Column 8 of A187783.
Number of permutations of a multiset that contains n different elements, each occurring  8 times.
Or in other words (the former title of this sequence):
Number of 8*n X n 0..1 arrays with row sums 1 and column sums 8.
(End)

			a(3) = (8*3)!/(8!^3) = 9465511770 is the number of permutations of a multiset that contains 3 different elements 8 times, e.g., {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}.

Tilman Piesk, Table of n, a(n) for n = 0..54 (first 12 terms from R. H. Hardin)

[Factorial(8*n)/(40320^n):  n in [0..20]]; // Vincenzo Librandi, Nov 01 2014

A172609:=n->(8*n)!/(40320^n): seq(A172609(n), n=0..10); # Wesley Ivan Hurt, Nov 01 2014

Table[(8 n)! / (40320^n), {n, 0, 10}] (* Vincenzo Librandi, Nov 01 2014 *)

a(n) = (8n)!/(8!^n).

Name changed by Tilman Piesk, Oct 30 2014

A172613 a(n) = (9n)!/(9!^n).

Original entry on oeis.org

1, 1, 48620, 227873431500, 21452752266265320000, 19010638202652030712978200000, 101097362223624462291180422369532000000, 2392741010223442438553822446842770682716580000000, 203653377853981828616656775313699668953042169048889600000000

Offset: 0

R. H. Hardin, Feb 06 2010

From Tilman Piesk, Oct 30 2014: (Start)
Column 9 of A187783.
Number of permutations of a multiset that contains n different elements, each occurring 9 times.
Or in other words (the former title of this sequence):
Number of 9*n X n 0..1 arrays with row sums 1 and column sums 9.
(End)

			a(3) = (9*3)!/(9!^3) = 227873431500 is the number of permutations of a multiset that contains 3 different elements, each occurring 9 times, e.g., {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}.

Tilman Piesk, Table of n, a(n) for n = 0..54 (first 11 terms from R. H. Hardin)

[Factorial(9*n)/(362880^n): n in [0..20]]; // Vincenzo Librandi, Nov 01 2014

A172613:=n->(9*n)!/(362880^n): seq(A172613(n), n=0..10); # Wesley Ivan Hurt, Nov 01 2014

Table[(9 n)! / (362880^n), {n, 0, 10}] (* Vincenzo Librandi, Nov 01 2014 *)

a(n) = (9n)!/(9!^n).

Name changed by Tilman Piesk, Oct 30 2014

A347811 Number A(n,k) of k-dimensional lattice walks from {n}^k to {0}^k using steps that decrease the Euclidean distance to the origin and that change each coordinate by at most 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 19, 25, 1, 1, 1, 323, 211075, 241, 1, 1, 1, 38716, 1322634996717, 2062017739, 2545, 1, 1, 1, 32253681, 16042961630858858915656, 29261778984922904560001, 32191353922714, 28203, 1, 1

Offset: 0

Alois P. Heinz, Sep 14 2021

Lattice points may have negative coordinates, and different walks may differ in length. All walks are self-avoiding.

			Square array A(n,k) begins:
  1, 1,     1,              1,                       1,     1, ...
  1, 1,     3,             19,                     323, 38716, ...
  1, 1,    25,         211075,           1322634996717, ...
  1, 1,   241,     2062017739, 29261778984922904560001, ...
  1, 1,  2545, 32191353922714, ...
  1, 1, 28203, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..11, flattened
Alois P. Heinz, Animation of A(2,2) = 25 walks
Wikipedia, Counting lattice paths
Wikipedia, Self-avoiding walk

Columns k=0+1, 2-3 give: A000012, A346539, A347813.
Rows n=0-2 give: A000012, A346840, A347812.
Main diagonal gives A347810.
Cf. A089759, A187783, A335570.

s:= proc(n) option remember;
     `if`(n=0, [[]], map(x-> seq([x[], i], i=-1..1), s(n-1)))
    end:
b:= proc(l) option remember; (n-> `if`(l=[0$n], 1, add((h-> `if`(
      add(i^2, i=h) b([n$k]):
seq(seq(A(n, d-n), n=0..d), d=0..8);

s[n_] := s[n] = If[n == 0, {{}}, Sequence @@ Table[Append[#, i], {i, -1, 1}]& /@ s[n-1]];
b[l_List] := b[l] = With[{n = Length[l]}, If[l == Table[0, {n}], 1, Sum[With[{h = l+x}, If[h.h < l.l, b[Sort[h]], 0]], {x, s[n]}]]];
A[n_, k_] := b[Table[n, {k}]];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Nov 03 2021, after Alois P. Heinz *)

A120666 Triangle read by rows: T(n, k) = (n*k)!/(n!)^k.

Original entry on oeis.org

1, 1, 6, 1, 20, 1680, 1, 70, 34650, 63063000, 1, 252, 756756, 11732745024, 623360743125120, 1, 924, 17153136, 2308743493056, 1370874167589326400, 2670177736637149247308800, 1, 3432, 399072960, 472518347558400, 3177459078523411968000, 85722533226982363751829504000, 7363615666157189603982585462030336000

Offset: 1

Roger L. Bagula, Aug 11 2006

T(m,n) is the number of ways to distribute n*m different toys among m different kids so that each kid gets exactly n toys. For example, with n=3 and m=2, the 6 different toys, t1, t2, t3, t4, t5 and t6, can be distributed in exactly 20 ways among the 2 kids, k1 and k2, since there are C(6,3)=20 ways to choose the three toys for k1, with the other three toys going to k2. The proof for the general case is based on the identity C(n*m,n)*C(n*m-n,n)*...*C(n*m-n*(m-1),n) = (n*m)!/(n!)^m. - Dennis P. Walsh, Apr 12 2018

			Triangle begins:
  1;
  1,   6;
  1,  20,   1680;
  1,  70,  34650,    63063000;
  1, 252, 756756, 11732745024, 623360743125120;

Seiichi Manyama, Rows n = 1..26, flattened
Eric Weisstein's World of Mathematics, Macdonald's Constant-Term Conjecture

Cf. A000984, A006480, A034841, A089759, A187783.

[Factorial(n*k)/(Factorial(n))^k: k in [1..n], n in [1..10]]; // G. C. Greubel, Dec 26 2022

T:= (m, n)-> (n*m)!/(m!)^n:
seq(seq(T(m, n), n=1..m), m=1..7);  # Alois P. Heinz, Apr 12 2018

Table[(n*k)!/(n!)^k, {n,10}, {k,n}]//Flatten

def A120666(n,k): return gamma(n*k+1)/(factorial(n))^k
flatten([[A120666(n,k) for k in range(1,n+1)] for n in range(1,11)]) # G. C. Greubel, Dec 26 2022

T(n, k) = (k*n)!/(n!)^k.

Edited by N. J. A. Sloane, Jun 17 2007
Offset corrected by Alois P. Heinz, Apr 12 2018
New name using formula by Joerg Arndt, Apr 15 2018

A141906 Triangle t(n,m) = (n*m)!/(m!^n) read by rows, 0<=m<=n.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 1, 6, 90, 1680, 1, 24, 2520, 369600, 63063000, 1, 120, 113400, 168168000, 305540235000, 623360743125120, 1, 720, 7484400, 137225088000, 3246670537110000, 88832646059788350720, 2670177736637149247308800

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 14 2008

Row sums are in A221177.

			1;
1, 1;
1, 2, 6;
1, 6, 90, 1680;
1, 24, 2520, 369600, 63063000;
1, 120, 113400, 168168000, 305540235000, 623360743125120;
1, 720, 7484400, 137225088000, 3246670537110000, 88832646059788350720, 2670177736637149247308800;

Seiichi Manyama, Rows n = 0..26, flattened

Cf. A034841, A187783, A221177.

A141906 := proc(n,m)
        (n*m)!/m!^n ;
end proc:
seq(seq(A141906(n,m),m=0..n),n=0..5) ; # R. J. Mathar, Nov 08 2011

Clear[t, n, m]; t[n_, m_] = (n*m)!/m!^n; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

A375694 Number A(n,k) of multiset permutations of {{1}^k, {2}^k, ..., {n}^k} with no fixed k-tuple {j}^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 5, 2, 0, 1, 0, 19, 74, 9, 0, 1, 0, 69, 1622, 2193, 44, 0, 1, 0, 251, 34442, 362997, 101644, 265, 0, 1, 0, 923, 756002, 62924817, 166336604, 6840085, 1854, 0, 1, 0, 3431, 17150366, 11729719509, 305225265804, 136221590695, 630985830, 14833, 0

Offset: 0

Alois P. Heinz, Aug 24 2024

			A(2,2) = 5: 1212, 1221, 2112, 2121, 2211.
A(2,3) = 19: 112122, 112212, 112221, 121122, 121212, 121221, 122112, 122121, 122211, 211122, 211212, 211221, 212112, 212121, 212211, 221112, 221121, 221211, 222111.
A(3,2) = 74: 121323, 121332, 122313, 122331, 123123, 123132, 123213, 123231, 123312, 123321, 131223, 131232, 131322, 132123, 132132, 132312, 132321, 133122, 133212, 133221, 211323, 211332, 212313, 212331, 213123, 213132, 213213, 213231, 213312, 213321, 221313, 221331, 223113, 223131, 223311, 231123, 231132, 231213, 231231, 231312, 231321, 232113, 232131, 232311, 233112, 233121, 233211, 311223, 311232, 311322, 312123, 312132, 312312, 312321, 313122, 313212, 313221, 321123, 321132, 321213, 321231, 321312, 321321, 322113, 322131, 322311, 323112, 323121, 323211, 331122, 331212, 331221, 332112, 332121.
A(4,1) = 9: 2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321.
Square array A(n,k) begins:
  1,  1,      1,         1,            1,               1, ...
  0,  0,      0,         0,            0,               0, ...
  0,  1,      5,        19,           69,             251, ...
  0,  2,     74,      1622,        34442,          756002, ...
  0,  9,   2193,    362997,     62924817,     11729719509, ...
  0, 44, 101644, 166336604, 305225265804, 623302086965044, ...

Alois P. Heinz, Antidiagonals n = 0..53, flattened

Columns k=0-2 give: A000007, A000166, A374980.
Rows n=0-2 give: A000012, A000004, A030662.
Main diagonal gives A375693.
Cf. A060538, A089759, A187783, A372307.

A:= (n, k)-> add((-1)^(n-j)*binomial(n, j)*(k*j)!/k!^j, j=0..n):
seq(seq(A(n, d-n), n=0..d), d=0..10);

A(n,k) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*(k*j)!/k!^j.

A249619 Triangle T(m,n) = number of permutations of a multiset with m elements and signature corresponding to n-th integer partition (A194602).

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 24, 12, 4, 6, 1, 120, 60, 20, 30, 5, 10, 1, 720, 360, 120, 180, 30, 60, 6, 90, 15, 20, 1, 5040, 2520, 840, 1260, 210, 420, 42, 630, 105, 140, 7, 210, 21, 35, 1, 40320, 20160, 6720, 10080, 1680, 3360, 336, 5040, 840, 1120, 56

Offset: 0

Tilman Piesk, Nov 04 2014

This triangle shows the same numbers in each row as A036038 and A078760 (the multinomial coefficients), but in this arrangement the multisets in column n correspond to the n-th integer partition in the infinite order defined by A194602.
Row lengths: A000041 (partition numbers), Row sums: A005651
Columns: 0: A000142 (factorials), 1: A001710, 2: A001715, 3: A133799, 4: A001720, 6: A001725, 10: A001730, 14: A049388
Last in row: end-2: A037955 after 1 term mismatch, end-1: A001405, end: A000012
The rightmost columns form the triangle A173333:
  n       0      1     2     4    6  10  14  21      (A000041(1,2,3...)-1)
m
1         1
2         2      1
3         6      3     1
4        24     12     4     1
5       120     60    20     5    1
6       720    360   120    30    6   1
7      5040   2520   840   210   42   7   1
8     40320  20160  6720  1680  336  56   8   1
A249620 shows the number of partitions of the same multisets. A187783 shows the number of permutations of special multisets.

			Triangle begins:
  n     0    1    2    3   4   5  6   7   8   9 10
m
0       1
1       1
2       2    1
3       6    3    1
4      24   12    4    6   1
5     120   60   20   30   5  10  1
6     720  360  120  180  30  60  6  90  15  20  1

Tilman Piesk, Triangle rows m=0..25, flattened
Tilman Piesk, Illustration of the multisets for m=0..6

Cf. A036038, A078760, A005651, A005651, A249620, A000041, A173333.

A198801 Number of closed paths of length 19n whose steps are 19th roots of unity.

Original entry on oeis.org

1, 121645100408832000, 997586474354936812896742294502400000000, 66507379539349211518364492838558005847493108039680000000000000, 11256716378122801825351385824819232115042452248916289339300576523719750000000000000000

Offset: 0

Simon Plouffe, Oct 30 2011

nonn

Equivalently, the number of paths of length 19n in Z^19 from {0}^19 to {n}^19. - Andrew Howroyd, Nov 01 2018

Andrew Howroyd, Table of n, a(n) for n = 0..40
Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.

Row n=19 of A187783, column k=19 of A089759.

a(n) = (19*n)!/(n!)^19; \\ Andrew Howroyd, Nov 01 2018

a(n) = (19*n)!/(n!)^19. - Andrew Howroyd, Nov 01 2018

Sequence redefined and a(2)-a(4) from Andrew Howroyd, Nov 01 2018

cp's OEIS Frontend

A248814 a(n) = (6n)!/(6!^n).

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A172603 a(n) = (7n)!/(7!^n).

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A172609 a(n) = (8n)!/(8!^n).

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A172613 a(n) = (9n)!/(9!^n).

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A347811 Number A(n,k) of k-dimensional lattice walks from {n}^k to {0}^k using steps that decrease the Euclidean distance to the origin and that change each coordinate by at most 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A120666 Triangle read by rows: T(n, k) = (n*k)!/(n!)^k.

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A141906 Triangle t(n,m) = (n*m)!/(m!^n) read by rows, 0<=m<=n.