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 A060540 #106 Jul 30 2025 11:12:53
%S A060540 1,1,1,1,3,1,1,10,15,1,1,35,280,105,1,1,126,5775,15400,945,1,1,462,
%T A060540 126126,2627625,1401400,10395,1,1,1716,2858856,488864376,2546168625,
%U A060540 190590400,135135,1,1,6435,66512160,96197645544,5194672859376,4509264634875,36212176000,2027025,1
%N A060540 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 A060540 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 A060540 Seiichi Manyama, <a href="/A060540/b060540.txt">Antidiagonals n = 1..50, flattened</a> (first 20 antidiagonals from Harry J. Smith)
%H A060540 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 A060540 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.
%H A060540 Elena L. Wang and Guoce Xin, <a href="https://arxiv.org/abs/2507.15654">On Ward Numbers and Increasing Schröder Trees</a>, arXiv:2507.15654 [math.CO], 2025. See p. 13.
%F A060540 T(n,k) = (n*k)!/(k!^n*n!) = T(n-1,k)*A060543(n,k) = A060538(n,k)/k!.
%F A060540 T(n,k) = Product_{j=2..n} binomial(j*k-1,k-1). - _M. F. Hasler_, Aug 22 2014
%e A060540 Array begins:
%e A060540 1, 1, 1, 1, 1, 1, ...
%e A060540 1, 3, 10, 35, 126, 462, ...
%e A060540 1, 15, 280, 5775, 126126, 2858856, ...
%e A060540 1, 105, 15400, 2627625, 488864376, 96197645544, ...
%e A060540 1, 945, 1401400, 2546168625, 5194672859376, 11423951396577720, ...
%e A060540 ...
%t A060540 T[n_, k_] := (n*k)!/(k!^n*n!);
%t A060540 Table[T[n-k+1, k], {n, 1, 10}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Jun 29 2018 *)
%o A060540 (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 A060540 Rows include A000012, A001700, A060542, A082368, A322252.
%Y A060540 Columns k=1..10 give A000012, A001147, A025035, A025036, A025037, A025038, A025040, A025041, A025042.
%Y A060540 Main diagonal is A057599.
%Y A060540 Related to A057599, see also A096126 and A246048.
%Y A060540 Cf. A060358, A361948 (includes row/col 0).
%Y A060540 Cf. A000217, A000292, A000332, A000389, A000579, A000580, A007318, A036040, A099174, A133314, A132440, A135278 (associations in Copeland link).
%K A060540 nonn,tabl,easy
%O A060540 1,5
%A A060540 _Henry Bottomley_, Apr 02 2001
%E A060540 Definition reworded by _M. F. Hasler_, Aug 23 2014