A071551 -id:A071551 - 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

A008979 a(n) = (6n)!/(n!)^6.

Original entry on oeis.org

1, 720, 7484400, 137225088000, 3246670537110000, 88832646059788350720, 2670177736637149247308800, 85722533226982363751829504000, 2889253496242619386328267523990000, 101097362223624462291180422369532000000, 3644153415887633116359073848179365185734400, 134567406165969006655507763343147223231094784000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..100
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Cf. A000897, A000984, A002894, A002897, A006480, A008977, A008978, A071549, A071550, A071551, A071552, A186420, A188662.

[Factorial(6*n)/Factorial(n)^6: n in [0..20]]; // Vincenzo Librandi, Aug 13 2014

seq( (6*n)!/(n!)^6, n=0..20); # G. C. Greubel, Feb 17 2020

Table[(6 n)!/(n)!^6, {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2014 *)

vector(21, n, my(m=n-1); (6*m)!/(m!)^6 ) \\ G. C. Greubel, Feb 17 2020

[factorial(6*n)/factorial(n)^6 for n in (0..20)] # G. C. Greubel, Feb 17 2020

From Peter Bala, Jul 12 2016: (Start)
a(n) = binomial(2*n,n)*binomial(3*n,n)*binomial(4*n,n)*
  binomial(5*n,n)*binomial(6*n,n) = ( [x^n](1 + x)^(2*n) ) * ( [x^n](1 + x)^(3*n) ) * ( [x^n](1 + x)^(4*n) ) * ( [x^n](1 + x)^(5*n) ) * ( [x^n](1 + x)^(6*n) ) = [x^n](F(x)^(720*n)), where F(x) = 1 + x + 4478*x^2 + 53085611*x^3 + 926072057094*x^4 + 19977558181209910*x^5 + 493286693783478576177*x^6 + ... appears to have integer coefficients. For similar results see A000897, A002894, A002897, A006480, A008977, A008978, A186420 and A188662. (End)
a(n) ~ 3^(6*n+1/2)*4^(3*n-1)/(Pi*n)^(5/2). - Ilya Gutkovskiy, Jul 12 2016
From Peter Bala, Feb 14 2020: (Start)
a(m*p^k) == a(m*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers m and k - apply Mestrovic, equation 39, p. 12.
a(n) = [(x*y*z*u*v)^n] (1 + x + y + z + u + v)^(6*n). (End)

A071549 a(n) = (7n)!/n!^7.

Original entry on oeis.org

1, 5040, 681080400, 182509367040000, 66475579247327250000, 28837919555681211870935040, 14007180988362844601443040716800, 7363615666157189603982585462030336000, 4104167472585675600759440022842715359250000, 2392741010223442438553822446842770682716580000000

Offset: 0

Benoit Cloitre, May 30 2002

nonn

Number of closed paths of length 7n whose steps are 7th roots of unity. - Andrew Howroyd, Nov 01 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Row n=7 of A187783, column k=7 of A089759.

Sequences (k*n)!/n!^k: A000984 (k = 2), A006480 (k =3), A008977 (k = 4), A008978 (k = 5), A008979 (k = 6), A071550 (k = 8), A071551 (k = 9), A071552 (k = 10).

[Factorial(7*n)/Factorial(n)^7: n in [0..20]]; // Vincenzo Librandi, Aug 13 2014

Table[(7 n)!/(n)!^7, {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2014 *)

a(n) = (7*n)!/(n!^7); \\ Andrew Howroyd, Nov 01 2018

From Peter Bala, Feb 14 2020: (Start)
a(n) = C(7*n,n)*C(6*n,n)*C(5*n,n)*C(4*n,n)*C(3*n,n)*C(2*n,n).
a(m*p^k) == a(m*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers m and k - apply  Mestrovic, Equation 39, p. 12.
a(n) = [x^n](F(x)^(5040*n)), where F(x) = 1 + x + 62528*x^2 + 11087269661*x^3 + 3021437267047869*x^4 + 1045823730475703710735*x^5 + ...
appears to have integer coefficients. For similar results see A008979.
a(n) = [(x*y*z*u*v*w)^n] (1 + x + y + z + u + v + w)^(7*n). (End)

a(8)-a(9) added by Andrew Howroyd, Nov 01 2018

A071550 a(n) = (8n)!/n!^8.

Original entry on oeis.org

1, 40320, 81729648000, 369398958888960000, 2390461829733887910000000, 18975581770994682860770223800320, 171889289584866507880743491472699801600

Offset: 0

Benoit Cloitre, May 30 2002

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Sequences (k*n)!/n!^k: A000984 (k = 2), A006480 (k = 3), A008977 (k = 4), A008978 (k = 5), A008979 (k = 6), A071549 (k = 7), A071551 (k = 9), A071552 (k = 10).

[Factorial(8*n)/Factorial(n)^8: n in [0..20]]; // Vincenzo Librandi, Aug 13 2014

Table[(8 n)!/(n)!^8, {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2014 *)

A071552 a(n) = (10n)!/n!^10.

Original entry on oeis.org

1, 3628800, 2375880867360000, 4386797336285844480000000, 12868639981414579848070084500000000, 49120458506088132224064306071170476903628800

Offset: 0

Benoit Cloitre, May 30 2002

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Sequences (k*n)!/n!^k: A000984 (k = 2), A006480 (k = 3), A008977 (k = 4), A008978 (k = 5), A008979 (k = 6), A071549 (k = 7), A071550 (k = 8), A071551 (k = 9).

[Factorial(10*n)/Factorial(n)^10: n in [0..20]]; // Vincenzo Librandi, Aug 13 2014

Table[(10n)!/(n)!^10, {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2014 *)

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

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A008979 a(n) = (6n)!/(n!)^6.

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A071549 a(n) = (7n)!/n!^7.

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A071550 a(n) = (8n)!/n!^8.

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A071552 a(n) = (10n)!/n!^10.

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A269127 Number of sequences with n copies each of 1,2,...,9 avoiding the pattern 12...9.

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