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 A053529 #61 Jan 05 2025 00:45:02
%S A053529 1,1,4,18,120,840,7920,75600,887040,10886400,152409600,2235340800,
%T A053529 36883123200,628929100800,11769069312000,230150688768000,
%U A053529 4833164464128000,105639166144512000,2464913876705280000,59606099200327680000,1525429559126753280000,40464026199993876480000
%N A053529 a(n) = n! * number of partitions of n.
%C A053529 Commuting permutations: number of ordered pairs (g, h) in Sym(n) such that gh = hg.
%C A053529 Equivalently sum of the order of all normalizers of all cyclic subgroups of Sym(n). - _Olivier Gérard_, Apr 04 2012
%C A053529 From _Gus Wiseman_, Jan 16 2019: (Start)
%C A053529 Also the number of Young tableaux with distinct entries from 1 to n, where a Young tableau is an array obtained by replacing the dots in the Ferrers diagram of an integer partition of n with positive integers. For example, the a(3) = 18 tableaux are:
%C A053529 123 213 132 312 231 321
%C A053529 .
%C A053529 12 21 13 31 23 32
%C A053529 3 3 2 2 1 1
%C A053529 .
%C A053529 1 2 1 3 2 3
%C A053529 2 1 3 1 3 2
%C A053529 3 3 2 2 1 1
%C A053529 (End)
%D A053529 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.12, solution.
%H A053529 T. D. Noe, <a href="/A053529/b053529.txt">Table of n, a(n) for n = 0..200</a>
%H A053529 M. Holloway, M. Shattuck, <a href="http://puma.dimai.unifi.it/24_1/2.holloway_shattuck.pdf">Commuting pairs of functions on a finite set</a>, PU.M.A, Volume 24 (2013), Issue No. 1.
%H A053529 M. Holloway, M. Shattuck, <a href="http://www.researchgate.net/profile/Mark_Shattuck/publication/272492907">Commuting pairs of functions on a finite set</a>, Research Gate, 2015.
%H A053529 R. P. Stanley, <a href="http://www.jstor.org/stable/2589191">Pairs with equal squares, Problem 10654</a>, Amer. Math. Monthly, 107 (April 2000), solution p. 368.
%H A053529 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F A053529 E.g.f: Sum_{n>=0} x^n/(Product_{k=1..n} 1-x^k) = exp(Sum_{n>=1} (x^n/n)/(1-x^n)). - _Joerg Arndt_, Jan 29 2011
%F A053529 a(n) = Sum{k=1..n} (((n-1)!/(n-k)!)*sigma(k)*a(n-k)), n > 0, and a(0)=1. See A274760. - _Johannes W. Meijer_, Jul 28 2016
%F A053529 a(n) ~ sqrt(Pi/6)*exp(sqrt(2/3)*Pi*sqrt(n))*n^n/(2*exp(n)*sqrt(n)). - _Ilya Gutkovskiy_, Jul 28 2016
%p A053529 seq(count(Permutation(n))*count(Partition(n)),n=1..20); # _Zerinvary Lajos_, Oct 16 2006
%p A053529 with(combinat): A053529 := proc(n): n! * numbpart(n) end: seq(A053529(n), n=0..20); # _Johannes W. Meijer_, Jul 28 2016
%t A053529 Table[PartitionsP[n] n!, {n, 0, 20}] (* _T. D. Noe_, Jun 19 2012 *)
%o A053529 (PARI) N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, x^k/(1-x^k)/k)))) \\ _Joerg Arndt_, Apr 16 2010
%o A053529 (PARI) N=66; x='x+O('x^N); Vec(serlaplace(sum(n=0, N, x^n/prod(k=1,n,1-x^k)))) \\ _Joerg Arndt_, Jan 29 2011
%o A053529 (PARI) a(n) = n!*numbpart(n); \\ _Michel Marcus_, Jul 28 2016
%o A053529 (Magma) a:= func< n | NumberOfPartitions(n)*Factorial(n) >; [ a(n) : n in [0..25]]; // _Vincenzo Librandi_, Jan 17 2019
%o A053529 (Python)
%o A053529 from math import factorial
%o A053529 from sympy import npartitions
%o A053529 def A053529(n): return factorial(n)*npartitions(n) # _Chai Wah Wu_, Jul 10 2023
%Y A053529 Column k=2 of A362827.
%Y A053529 Cf. A000041, A072169, A061256.
%Y A053529 Sequences counting pairs of functions from an n-set to itself: A053529, A181162, A239749-A239785, A239836-A239841.
%Y A053529 Cf. A000085, A117433, A153452, A296188, A323295, A323434, A323436.
%K A053529 nonn,easy,nice
%O A053529 0,3
%A A053529 _N. J. A. Sloane_, Jan 16 2000