%I #49 Jun 26 2024 06:25:15
%S 1,2,7,52,749,17686,614227,29354312,1844279257,147273109354,
%T 14561325802271,1745720380045852,249461639720702917,
%U 41886684733511640062,8164388189339113521259,1828191138807263097870256,466057478369217965809683377,134193343258948416556377786322
%N Sum of mistyped version of binomial coefficients.
%C 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 GĂ©rard_, Mar 05 2024
%C Binomial transform of (n!)^2. - _Peter Luschny_, May 31 2014
%H Seiichi Manyama, <a href="/A046662/b046662.txt">Table of n, a(n) for n = 0..253</a>
%H Roland Bacher, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i3p7">Counting Packings of Generic Subsets in Finite Groups</a>, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
%F a(n) = Sum_{k=0..n} n!*k!/(n-k)!.
%F E.g.f.: exp(x)*F(x), with F(x) = Sum_{k>=0} k!*x^k. - _Ralf Stephan_, Apr 02 2004
%F a(n) = n^2*a(n - 1) - n*(n - 1)*a(n - 2) + 1. - _Vladeta Jovovic_, Jul 15 2004
%F From _Peter Bala_, Nov 26 2017: (Start)
%F a(k) == a(0) (mod k) for all k (by the inhomogeneous recurrence equation).
%F 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 G.f.: Sum_{k>=0} (k!)^2*x^k/(1 - x)^(k+1). - _Ilya Gutkovskiy_, Apr 12 2019
%F a(n) ~ (n!)^2. - _Vaclav Kotesovec_, May 03 2021
%F a(n) = 3F0(1,1,-n;;-1). - _R. J. Mathar_, Jun 26 2024
%t Table[Sum[(n!k!)/(n-k)!,{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Sep 29 2012 *)
%Y Cf. A003149, A064570, A229464.
%Y Cf. A000522 (Total number of ordered k-subsets of [1,n], k=0..n.)
%K nonn,easy
%O 0,2
%A _Len Smiley_
%E Corrected and extended by _Harvey P. Dale_, Sep 29 2012