This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A187783 #68 Mar 23 2025 18:25:25
%S A187783 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,
%T A187783 1,1,252,34650,369600,113400,720,1,1,1,924,756756,63063000,168168000,
%U A187783 7484400,5040,1
%N A187783 De Bruijn's triangle, T(m,n) = (m*n)!/(n!^m) read by downward antidiagonals.
%C A187783 From _Tilman Piesk_, Oct 28 2014: (Start)
%C A187783 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.
%C A187783 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:
%C A187783 m\n:0 1 2 3 4 5
%C A187783 0: 1 1 1 1 1 1 ... A000012;
%C A187783 1: 1 1 1 1 1 1 ... A000012;
%C A187783 2: 1 2 6 20 70 252 ... A000984;
%C A187783 3: 1 6 90 1680 34650 756756 ... A006480;
%C A187783 4: 1 24 2520 369600 63063000 11732745024 ... A008977;
%C A187783 5: 1 120 113400 168168000 305540235000 623360743125120 ... A008978;
%C A187783 6: 1 720 7484400 137225088000 3246670537110000 88832646059788350720 ... A008979;
%C A187783 with columns: A000142 (n=1), A000680 (n=2), A014606 (n=3), A014608 (n=4), A014609 (n=5).
%C A187783 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).
%C A187783 (End)
%H A187783 Tilman Piesk, <a href="/A187783/b187783.txt">First 54 rows of the triangle, flattened</a>
%H A187783 T. Chappell, A. Lascoux, S. Ole Warnaar, and W. Zudilin, <a href="http://arxiv.org/abs/1112.3130">Logarithmic and complex constant term identities</a>, arXiv:1112.3130 [math.CO], 2012.
%H A187783 Tilman Piesk, <a href="/A187783/a187783.txt">Array for indices 0..16</a>
%H A187783 Tilman Piesk, <a href="http://pastebin.com/4j8qqrff">PHP code used to create the b-file</a>
%H A187783 Tilman Piesk, <a href="https://commons.wikimedia.org/wiki/File:Multisets_that_contain_m_different_elements_n_times.svg">Illustration of the multisets for m,n=0..4</a>
%H A187783 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation#Permutations_of_multisets">Permutations of multisets</a>, <a href="http://en.wikipedia.org/wiki/Pascal_matrix">Pascal matrix</a> and <a href="http://en.wikipedia.org/wiki/Pascal%27s_simplex">simplex</a>
%F A187783 T(m,n) = (m*n)!/(n!)^m.
%F A187783 A060540(m,n) = T(m,n)/m! . - _R. J. Mathar_, Jun 21 2023
%e A187783 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}.
%t A187783 T[n_, k_]:= (k*n)!/(n!)^k; Table[T[n, k-n], {k, 9}, {n, 0, k-1}]//Flatten
%o A187783 (Magma) [Factorial(k*(n-k))/(Factorial(n-k))^k: k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 26 2022
%o A187783 (SageMath)
%o A187783 def A187783(n,k): return gamma(k*(n-k)+1)/(factorial(n-k))^k
%o A187783 flatten([[A187783(n,k) for k in range(n+1)] for n in range(11)]) # _G. C. Greubel_, Dec 26 2022
%Y A187783 Cf. A089759 (transposed), A141906 (subtriangle), A120666 (subtriangle transposed), A060538 (1st row/column removed).
%Y A187783 Rows: A000012, A000984, A006480, A006480, A008978, A008979, A071549, A071550, A071551, A071552.
%Y A187783 Columns: A000012, A000142, A000680, A014606, A014608, A014609, A248814, A172603, A172609, A172613.
%Y A187783 Main diagonal gives: A034841.
%Y A187783 Row sums of the triangle: A248827.
%K A187783 nonn,tabl,easy
%O A187783 0,9
%A A187783 _Robert G. Wilson v_, Jan 05 2013
%E A187783 Row m=0 prepended by _Tilman Piesk_, Oct 28 2014