A014609 -id:A014609 - cp's OEIS frontend

A187783 De Bruijn's triangle, T(m,n) = (m*n)!/(n!^m) read by downward antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 6, 1, 1, 1, 20, 90, 24, 1, 1, 1, 70, 1680, 2520, 120, 1, 1, 1, 252, 34650, 369600, 113400, 720, 1, 1, 1, 924, 756756, 63063000, 168168000, 7484400, 5040, 1

Offset: 0

Robert G. Wilson v, Jan 05 2013

From Tilman Piesk, Oct 28 2014: (Start)
Number of permutations of a multiset that contains m different elements n times. These multisets have the signatures A249543(m,n-1) for m>=1 and n>=2.
In an m-dimensional Pascal tensor (the generalization of a symmetric Pascal matrix) P(x1,...,xn) = (x1+...+xn)!/(x1!*...*xn!), so the main diagonal of an m-dimensional Pascal tensor is D(n) = (m*n)!/(n!^m). These diagonals are the rows of this array (with m>0), which begins like this:
  m\n:0   1       2            3                4                    5
  0:  1   1       1            1                1                    1 ... A000012;
  1:  1   1       1            1                1                    1 ... A000012;
  2:  1   2       6           20               70                  252 ... A000984;
  3:  1   6      90         1680            34650               756756 ... A006480;
  4:  1  24    2520       369600         63063000          11732745024 ... A008977;
  5:  1 120  113400    168168000     305540235000      623360743125120 ... A008978;
  6:  1 720 7484400 137225088000 3246670537110000 88832646059788350720 ... A008979;
with columns: A000142 (n=1), A000680 (n=2), A014606 (n=3), A014608 (n=4), A014609 (n=5).
A089759 is the transpose of this matrix. A034841 is its diagonal. A141906 is its lower triangle. A120666 is the upper triangle of this matrix with indices starting from 1. A248827 are the diagonal sums (or the row sums of the triangle).
(End)

			T(3,5) = (3*5)!/(5!^3) = 756756 = A014609(3) = A006480(5) is the number of permutations of a multiset that contains 3 different elements 5 times, e.g., {1,1,1,1,1,2,2,2,2,2,3,3,3,3,3}.

Tilman Piesk, First 54 rows of the triangle, flattened
T. Chappell, A. Lascoux, S. Ole Warnaar, and W. Zudilin, Logarithmic and complex constant term identities, arXiv:1112.3130 [math.CO], 2012.
Tilman Piesk, Array for indices 0..16
Tilman Piesk, PHP code used to create the b-file
Tilman Piesk, Illustration of the multisets for m,n=0..4
Wikipedia, Permutations of multisets, Pascal matrix and simplex

Cf. A089759 (transposed), A141906 (subtriangle), A120666 (subtriangle transposed), A060538 (1st row/column removed).
Rows: A000012, A000984, A006480, A006480, A008978, A008979, A071549, A071550, A071551, A071552.
Columns: A000012, A000142, A000680, A014606, A014608, A014609, A248814, A172603, A172609, A172613.
Main diagonal gives: A034841.
Row sums of the triangle: A248827.

[Factorial(k*(n-k))/(Factorial(n-k))^k: k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 26 2022

T[n_, k_]:= (k*n)!/(n!)^k; Table[T[n, k-n], {k, 9}, {n, 0, k-1}]//Flatten

def A187783(n,k): return gamma(k*(n-k)+1)/(factorial(n-k))^k
flatten([[A187783(n,k) for k in range(n+1)] for n in range(11)]) # G. C. Greubel, Dec 26 2022

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

A060540(m,n) = T(m,n)/m! . - R. J. Mathar, Jun 21 2023

Row m=0 prepended by Tilman Piesk, Oct 28 2014

A089759 Table T(n,k), 0<=k, 0<=n, read by antidiagonals, defined by T(n,k) = (k*n)! / (n!)^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 6, 1, 1, 1, 24, 90, 20, 1, 1, 1, 120, 2520, 1680, 70, 1, 1, 1, 720, 113400, 369600, 34650, 252, 1, 1, 1, 5040, 7484400, 168168000, 63063000, 756756, 924, 1, 1, 1, 40320, 681080400, 137225088000, 305540235000, 11732745024, 17153136, 3432, 1, 1

Offset: 0

Philippe Deléham, Jan 08 2004; revised Jun 08 2005

T(n,k) is the number of lattice paths from {n}^k to {0}^k using steps that decrement one component by 1. - Alois P. Heinz, May 06 2013

			Row n=0: 1, 1,   1,      1,           1,               1, ... A000012
Row n=1: 1, 1,   2,      6,          24,             120, ... A000142
Row n=2: 1, 1,   6,     90,        2520,          113400, ... A000680
Row n=3: 1, 1,  20,   1680,      369600,       168168000, ... A014606
Row n=4: 1, 1,  70,  34650,    63063000,    305540235000, ... A014608
Row n=5: 1, 1, 252, 756756, 11732745024, 623360743125120, ... A014609

Alois P. Heinz, Antidiagonals n = 0..32, flattened
T. Chappell, A. Lascoux, S. Ole Warnaar, W. Zudilin, Logarithmic and complex constant term identities, arXiv:1112.3130 [math.CO], 2012.

Cf. A000680, A014606, A014608, A014609, A000984, A187783 (transposed version).
Main diagonal gives A034841.
Cf. A210472, A225094.

T:= (n, k)-> (k*n)!/(n!)^k:
seq(seq(T(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Aug 16 2012

T[n_, k_] := (k*n)!/(n!)^k; Table[T[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 19 2015 *)

Corrected by Alois P. Heinz, Aug 16 2012

A237202 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} (-1)^i * binomial(5n+1,i) binomial(k+5-i,5)^n, 0 <= k <= 5*(n-1).

Original entry on oeis.org

1, 1, 25, 100, 100, 25, 1, 1, 200, 5925, 52800, 182700, 273504, 182700, 52800, 5925, 200, 1, 1, 1275, 167475, 6021225, 84646275, 554083761, 1858142825, 3363309675, 3363309675, 1858142825, 554083761, 84646275

Offset: 1

Yahia Kahloune, Feb 05 2014

In general, define b(k,e,p) = Sum_{i=0..k} (-1)^i*binomial(e*p+1,i)*binomial(k+e-i,e)^p. Then T(n,k) = b(k,5,n).
Using these coefficients we can obtain formulas for binomial(n,e)^p and for Sum_{i=1..n} binomial(e-1+i,e)^p.
In particular:
  binomial(n, e)^p = Sum_{k=0..e*(p-1)} b(k,e,p) * binomial(n+k, e*p).
  Sum_{i=1..n} binomial(e-1+i, e)^p = Sum_{k=0..e*(p-1)} b(k,e,p) * binomial(n+e+k, e*p+1).
T(n,k) is the number of permutations of 5 indistinguishable copies of 1..n with exactly k descents. A descent is a pair of adjacent elements with the second element less than the first. - Andrew Howroyd, May 08 2020

			T(n,0) = 1;
T(n,1) = 6^n - (5*n+1);
T(n,2) = 21^n - (5*n+1)*6^n + C(5*n+1,2);
T(n,3) = 56^n - (5*n+1)*21^n + C(5*n+1,2)*6^n - C(5*n+1,3) ;
T(n,4) = 126^n - (5*n+1)*56^n + C(5*n+1,2)*21^n - C(5*n+1,3)*6^n  + C(5*n+1,4).
Triangle T(n,k) begins:
1;
1, 25, 100, 100, 25, 1;
1, 200, 5925, 52800, 182700, 273504, 182700, 52800, 5925, 200, 1;
1, 1275, 167475, 6021225, 84646275, 554083761, 1858142825, 3363309675, 3363309675, 1858142825, 554083761, 84646275, 6021225, 167475, 125, 1;
1, 7750, 3882250, 447069750, 18746073375, 359033166276, 3575306548500, 20052364456500, 66640122159000, 135424590593500, 171219515211316, 135424590593500, 66640122159000, 20052364456500, 3575306548500, 359033166276, 18746073375, 447069750, 3882250, 7750, 1;
...
Example:
Sum_{i=1..n} C(4+i,5)^3 = C(n+5,16) + 200*C(n+6,16) + 5925*(n+7,16) + 52800*C(n+8,16) + 182700*C(n+9,16) + 273504*C(n+10,16) + 182700*C(n+11,16) + 52800*C(n+12,16) + 5925*C(n+13,16) + 200*C(n+14,16) + C(n+15,16).
C(n,5)^3 = C(n,15) + 200*C(n+1,15) + 5925*C(n+2,15) + 52800*C(n+3,15) + 182700*C(n+4,15) + 273504*C(n+5,15) + 182700*C(n+6,15) + 52800*C(n+7,15) + 5925*C(n+8,15) + 200*C(n+9,15) + C(n+10,15).

G. C. Greubel, Table of n, a(n) for the first 25 rows, flattened
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 12.

Columns k=2..5 are A151647, A151648, A151649, A151650.
Row sums are A014609.
Similar triangles for e=1..6: A173018 (or A008292), A154283, A174266, A236463, this sequence, A237252.
Sum_{i=1..n} binomial(4+i,5)^p for p=2..3 gives: A086025, A086026.
Cf. A087127, A087107, A087108, A087109, A087110, A087111, A181544.

b[k_, 5, p_] := Sum[(-1)^i*Binomial[5*p+1, i]*Binomial[k-i, 5]^p /. k -> 5+i, {i, 0, k-5}]; row[p_] := Table[b[k, 5, p], {k, 5, 5*p}]; Table[row[p], {p, 1, 5}] // Flatten (* Jean-François Alcover, Feb 05 2014 *)

T(n,k)={sum(i=0, k, (-1)^i*binomial(5*n+1, i)*binomial(k+5-i, 5)^n)} \\ Andrew Howroyd, May 08 2020

Sum_{i=1..n} binomial(4+i,5)^p = Sum{k=0..5*(p-1)} T(p,k) * binomial(n+5+k, 5*p+1).

binomial(n,5)^p = Sum_{k=0..5*(p-1)} T(p,k) * binomial(n+k, 5*p).

Edited by Andrew Howroyd, May 08 2020

A060538 Square array read by antidiagonals of number of ways of dividing n*k labeled items into n labeled boxes with k items in each box.

Original entry on oeis.org

1, 1, 2, 1, 6, 6, 1, 20, 90, 24, 1, 70, 1680, 2520, 120, 1, 252, 34650, 369600, 113400, 720, 1, 924, 756756, 63063000, 168168000, 7484400, 5040, 1, 3432, 17153136, 11732745024, 305540235000, 137225088000, 681080400, 40320, 1, 12870

Offset: 1

Henry Bottomley, Apr 02 2001

			       1        1        1        1
       2        6       20       70
       6       90     1680    34650
      24     2520   369600 63063000

Harry J. Smith, Table of n, a(n) for n = 1..210

Subtable of A187783.
Rows include A000012, A000984, A006480, A008977, A008978 etc.
Columns include A000142, A000680, A014606, A014608, A014609 etc.
Main diagonal is A034841.

T(n,k)=(n*k)!/k!^n;
for(n=1, 6, for(k=1, 6, print1(T(n,k), ", ")); print) \\ Harry J. Smith, Jul 06 2009

T(n, k) = (nk)!/k!^n = T(n-1, k)*binomial(nk, k) = T(n-1, k)*A060539(n, k) = A060540(n, k)*A000142(k).

A177301 Number of permutations of 5 copies of 1..n with all adjacent differences <= 1 in absolute value.

Original entry on oeis.org

1, 1, 252, 7002, 284002, 8956752, 276285002, 8039989002, 226901044252, 6232521421502, 167765555015002, 4441865811412752, 116042171156004002, 2998175631045512002, 76751828260736962252, 1949598912568702682502, 49197330547669842130002, 1234523426440511782592752

Offset: 0

R. H. Hardin, May 06 2010

nonn

a(n) = (5n)!/120^n = A014609(n) for n<=2.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A014609.

Column k=5 of A331562.

a(10) from Alois P. Heinz, Jan 21 2020

Terms a(11) and beyond from Andrew Howroyd, May 15 2020

A269115 Number of sequences with 5 copies each of 1,2,...,n avoiding the pattern 12...n.

Original entry on oeis.org

0, 0, 1, 26945, 1474050783, 173996758190594, 41489887418194798439, 18675035107760751336633904, 15006398706095549205738572240519, 20584293980310480336378076914369061769, 46436573450646384263588550418764923934014699

Offset: 0

Alois P. Heinz, Feb 19 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100

Column k=5 of A269129.

Cf. A014609, A268848.

a(n) = A014609(n) - A268848(n).

A177302 Number of permutations of 5 copies of 1..n with all adjacent differences <= 2 in absolute value.

Original entry on oeis.org

1, 1, 252, 756756, 457507512, 184619047020, 97087382008034, 49254897805981056, 22802419346927105088

Offset: 0

R. H. Hardin, May 06 2010

a(n) = (5n)!/120^n for n<=3.

Cf. A014609.

a(0), a(8) from Alois P. Heinz, Jan 19 2016

A177303 Number of permutations of 5 copies of 1..n with all adjacent differences <= 3 in absolute value.

Original entry on oeis.org

1, 1, 252, 756756, 11732745024, 52977847995072, 131319696504676680, 302061042944500510428, 847432462636241316703042

Offset: 0

R. H. Hardin, May 06 2010

nonn

a(n) = (5n)!/120^n for n<=4.

Cf. A014609.

a(0), a(8) from Alois P. Heinz, Jan 19 2016

cp's OEIS Frontend

A187783 De Bruijn's triangle, T(m,n) = (m*n)!/(n!^m) read by downward antidiagonals.

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A089759 Table T(n,k), 0<=k, 0<=n, read by antidiagonals, defined by T(n,k) = (k*n)! / (n!)^k.

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A237202 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} (-1)^i * binomial(5*n+1,i) * binomial(k+5-i,5)^n, 0 <= k <= 5*(n-1).

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A060538 Square array read by antidiagonals of number of ways of dividing n*k labeled items into n labeled boxes with k items in each box.

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A177301 Number of permutations of 5 copies of 1..n with all adjacent differences <= 1 in absolute value.

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A269115 Number of sequences with 5 copies each of 1,2,...,n avoiding the pattern 12...n.

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A177302 Number of permutations of 5 copies of 1..n with all adjacent differences <= 2 in absolute value.

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A177303 Number of permutations of 5 copies of 1..n with all adjacent differences <= 3 in absolute value.

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A237202 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} (-1)^i * binomial(5n+1,i) binomial(k+5-i,5)^n, 0 <= k <= 5*(n-1).