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%I #100 Feb 18 2024 10:02:24
%S 1,1,1,1,3,1,1,10,15,1,1,35,280,105,1,1,126,5775,15400,945,1,1,462,
%T 126126,2627625,1401400,10395,1,1,1716,2858856,488864376,2546168625,
%U 190590400,135135,1,1,6435,66512160,96197645544,5194672859376,4509264634875,36212176000,2027025,1
%N Square array read by antidiagonals downwards: T(n,k) = (n*k)!/(k!^n*n!), (n>=1, k>=1), the number of ways of dividing nk labeled items into n unlabeled boxes with k items in each box.
%C The Copeland link gives the associations of this entry with the operator calculus of Appell Sheffer polynomials, the combinatorics of simple set partitions encoded in the Faa di Bruno formula for composition of analytic functions (formal Taylor series), the Pascal matrix, and the geometry of the n-dimensional simplices (hypertriangles, or hypertetrahedra). These, in turn, are related to simple instances of the application of the exponential formula / principle / schema giving the number of not-necessarily-connected objects composed from an ensemble of connected objects. - _Tom Copeland_, Jun 09 2021
%H Seiichi Manyama, <a href="/A060540/b060540.txt">Antidiagonals n = 1..50, flattened</a> (first 20 antidiagonals from Harry J. Smith)
%H Tom Copeland, <a href="https://tcjpn.wordpress.com/2021/06/12/calculus-combinatorics-and-geometry-underlying-oeis-a060540-and-the-exponential-formula/">Calculus, Combinatorics, and Geometry Underlying OEIS A060540, and the Exponential Formula</a>, 2021.
%H Nattawut Phetmak and Jittat Fakcharoenphol, <a href="https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1555">Uniformly Generating Derangements with Fixed Number of Cycles in Polynomial Time</a>, Thai J. Math. (2023) Vol. 21, No. 4, 899-915. See pp. 901, 914.
%F T(n,k) = (n*k)!/(k!^n*n!) = T(n-1,k)*A060543(n,k) = A060538(n,k)/k!.
%F T(n,k) = Product_{j=2..n} binomial(j*k-1,k-1). - _M. F. Hasler_, Aug 22 2014
%e Array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 3, 10, 35, 126, 462, ...
%e 1, 15, 280, 5775, 126126, 2858856, ...
%e 1, 105, 15400, 2627625, 488864376, 96197645544, ...
%e 1, 945, 1401400, 2546168625, 5194672859376, 11423951396577720, ...
%e ...
%t T[n_, k_] := (n*k)!/(k!^n*n!);
%t Table[T[n-k+1, k], {n, 1, 10}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Jun 29 2018 *)
%o (PARI) { i=0; for (m=1, 20, for (n=1, m, k=m - n + 1; write("b060540.txt", i++, " ", (n*k)!/(k!^n*n!))); ) } \\ _Harry J. Smith_, Jul 06 2009
%Y Rows include A000012, A001700, A060542, A082368, A322252.
%Y Columns k=1..10 give A000012, A001147, A025035, A025036, A025037, A025038, A025040, A025041, A025042.
%Y Main diagonal is A057599.
%Y Related to A057599, see also A096126 and A246048.
%Y Cf. A060358, A361948 (includes row/col 0).
%Y Cf. A000217, A000292, A000332, A000389, A000579, A000580, A007318, A036040, A099174, A133314, A132440, A135278 (associations in Copeland link).
%K nonn,tabl,easy
%O 1,5
%A _Henry Bottomley_, Apr 02 2001
%E Definition reworded by _M. F. Hasler_, Aug 23 2014
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