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 A046662 #52 Apr 21 2025 13:24:17
%S A046662 1,2,7,52,749,17686,614227,29354312,1844279257,147273109354,
%T A046662 14561325802271,1745720380045852,249461639720702917,
%U A046662 41886684733511640062,8164388189339113521259,1828191138807263097870256,466057478369217965809683377,134193343258948416556377786322
%N A046662 Sum of mistyped version of binomial coefficients.
%C A046662 Origin of the name of this sequence: Binomial coefficients are n!/((n-k)!*k!) but if parentheses are omitted in the denominator, the formula might result in n!/(n-k)!*k! = n!*k!/(n-k)! and the sum giving a(n) instead of 2^n. If k! is forgotten altogether, one gets A000522. - _Olivier Gérard_, Mar 05 2024
%C A046662 Binomial transform of (n!)^2. - _Peter Luschny_, May 31 2014
%H A046662 Seiichi Manyama, <a href="/A046662/b046662.txt">Table of n, a(n) for n = 0..253</a>
%H A046662 Roland Bacher, <a href="https://doi.org/10.37236/2522">Counting Packings of Generic Subsets in Finite Groups</a>, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
%F A046662 a(n) = Sum_{k=0..n} n!*k!/(n-k)!.
%F A046662 E.g.f.: exp(x)*F(x), with F(x) = Sum_{k>=0} k!*x^k. - _Ralf Stephan_, Apr 02 2004
%F A046662 a(n) = n^2*a(n - 1) - n*(n - 1)*a(n - 2) + 1. - _Vladeta Jovovic_, Jul 15 2004
%F A046662 From _Peter Bala_, Nov 26 2017: (Start)
%F A046662 a(k) == a(0) (mod k) for all k (by the inhomogeneous recurrence equation).
%F A046662 More generally, a(n+k) = a(n) (mod k) for all n and k (by an induction argument on n). It follows that for each positive integer k, the sequence a(n) (mod k) is periodic, with the exact period dividing k. For example, modulo 10 the sequence becomes 1, 2, 7, 2, 9, 6, 7, 2, 7, 4, 1, 2, 7, 2, 9, 6, 7, 2, 7, 4, ... with exact period 10. (End)
%F A046662 G.f.: Sum_{k>=0} (k!)^2*x^k/(1 - x)^(k+1). - _Ilya Gutkovskiy_, Apr 12 2019
%F A046662 a(n) ~ (n!)^2. - _Vaclav Kotesovec_, May 03 2021
%F A046662 a(n) = 3F0(1,1,-n;;-1). - _R. J. Mathar_, Jun 26 2024
%t A046662 Table[Sum[(n!k!)/(n-k)!,{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Sep 29 2012 *)
%Y A046662 Cf. A003149, A064570, A229464.
%Y A046662 Cf. A000522 (Total number of ordered k-subsets of [1,n], k=0..n.)
%K A046662 nonn,easy
%O A046662 0,2
%A A046662 _Len Smiley_
%E A046662 Corrected and extended by _Harvey P. Dale_, Sep 29 2012