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A094258 a(1) = 1, a(n+1) = n*n! for n >= 1.

%I A094258 #51 Nov 28 2021 02:42:46
%S A094258 1,1,4,18,96,600,4320,35280,322560,3265920,36288000,439084800,
%T A094258 5748019200,80951270400,1220496076800,19615115520000,334764638208000,
%U A094258 6046686277632000,115242726703104000,2311256907767808000,48658040163532800000,1072909785605898240000,24728016011107368960000,594596384994354462720000
%N A094258 a(1) = 1, a(n+1) = n*n! for n >= 1.
%C A094258 The old definition was: "a(1) = 1, a(n+1) = n*(a(1) + a(2) + ... + a(n))."
%C A094258 a(n) is the number of positive integers k <= n! such that k is not divisible by n. It is also the number of rationals in (0,1] which can be written in the form m/n! but not in the form m/(n-1)!. - _Jonathan Sondow_, Aug 14 2006
%C A094258 Also, the number of monomials in the determinant of an n X n symbolic matrix with only one zero entry. The position of the zero in the matrix is not important. - _Artur Jasinski_, Apr 02 2008
%C A094258 From _Zak Seidov_, Jun 21 2005: (Start)
%C A094258 The number of integers that use each of the decimal digits 0 through n exactly once is the finite sequence 1, 1, 4, 18, 96, 600, 4320, 35280, 322560, 3265920, because there are (n+1)! permutations of decimal digits 0 through n, from which we remove the n! permutations with leading zero and get n*n! = total number of integers that use each of the decimal digits from 0 through n exactly once. For n=0 we have 1 integer (0) which uses zero exactly once, hence a(0)=1 by definition.
%C A094258 This sequence is finite because there are only 10 decimal digits. With the initial 1 replaced by 0, we get the initial terms of A001563, which is infinite. Cf. A109075 = number of primes which use each of the decimal digits from 0 through n exactly once. (End)
%C A094258 Partial sums yield factorial numbers A000142(n) = n! = (1, 2, 6, 24, 120, 720, ...). - _Vladimir Joseph Stephan Orlovsky_, Jun 27 2009
%H A094258 Harvey P. Dale, <a href="/A094258/b094258.txt">Table of n, a(n) for n = 1..400</a>
%H A094258 Jonathan Beagley and Lara Pudwell, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Pudwell/pudwell13.html">Colorful Tilings and Permutations</a>, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.
%H A094258 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H A094258 J. Sondow, <a href="https://www.jstor.org/stable/27642006">A geometric proof that e is irrational and a new measure of its irrationality</a>, Amer. Math. Monthly 113 (2006) 637-641.
%H A094258 J. Sondow, <a href="http://arxiv.org/abs/0704.1282">A geometric proof that e is irrational and a new measure of its irrationality</a>, arXiv:0704.1282 [math.HO], 2007-2010.
%F A094258 a(n+1) = n*n! = A001563(n) for n>=1.
%F A094258 From _Jonathan Sondow_, Aug 14 2006: (Start)
%F A094258 a(n) = n! - (n-1)! for n >= 2.
%F A094258 a(n) = n! - a(n-1) - a(n-2) - ... - a(1). with a(1) = 1. (End)
%F A094258 a(n) = A094304(n+1) = A001563(n-1) for n >= 2. - _Jaroslav Krizek_, Oct 16 2009
%F A094258 G.f.: 1/Q(0), where Q(k)= 1 + x/(1-x) - x/(1-x)*(k+2)/(1 - x/(1-x)*(k+1)/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Apr 22 2013
%F A094258 G.f.: W(0)*(1-sqrt(x)), where W(k) = 1 + sqrt(x)/( 1 - sqrt(x)*(k+2)/(sqrt(x)*(k+2) + 1/W(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 18 2013
%e A094258 a(1) = 1;
%e A094258 a(2) = 1*a(1) = 1;
%e A094258 ...
%e A094258 a(7) = 6*(a(1) + a(2) + ... + a(6)) = 6*(1 + 1 + 4 + 18 + 96 + 600) = 4320.
%p A094258 A094258 := proc(n) option remember: if n = 1 then 1; else (n-1)*add(A094258(i),i=1..n-1) ; fi ; end: seq(A094258(n),n=1..24) ; # _R. J. Mathar_, Jul 27 2007
%t A094258 a=s=1;lst={a};Do[a=s*n-s;s+=a;AppendTo[lst,a],{n,2,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Jun 27 2009 *)
%t A094258 Module[{lst={1}},Do[AppendTo[lst,n*Total[lst]],{n,30}];lst] (* _Harvey P. Dale_, Jul 01 2012 *)
%o A094258 (PARI) A094258(n)=(n-1)!*(n-(n>1)) \\ _M. F. Hasler_, Oct 21 2012
%Y A094258 Up to the offset and initial value, the same as A001563, cf. formula.
%Y A094258 Cf. A109075.
%K A094258 nonn
%O A094258 1,3
%A A094258 _Amarnath Murthy_, Apr 26 2004
%E A094258 Edited by Mark Hudson, Jan 05 2005
%E A094258 More terms from _R. J. Mathar_, Jul 27 2007
%E A094258 Edited by _M. F. Hasler_, Oct 21 2012
%E A094258 Edited by _Jon E. Schoenfield_, Jan 17 2015
%E A094258 Definition simplified by _M. F. Hasler_, Jun 28 2016