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A094258
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a(1) = 1, a(n+1) = n*( a(1) + a(2) + ... + a(n) ).
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7
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1, 1, 4, 18, 96, 600, 4320, 35280, 322560, 3265920, 36288000, 439084800, 5748019200, 80951270400, 1220496076800, 19615115520000, 334764638208000, 6046686277632000, 115242726703104000, 2311256907767808000, 48658040163532800000, 1072909785605898240000, 24728016011107368960000, 594596384994354462720000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| 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 (jsondow(AT)alumni.princeton.edu), Aug 14 2006
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 (grafix(AT)csl.pl), Apr 02 2008
Comment from Zak Seidov (zakseidov(AT)yahoo.com), Jun 21 2005: (Start) The number of integers which use each of the decimal digits 0 thrugh n exactly once is the finite sequence 1, 1, 4, 18, 96, 600, 4320, 35280, 322560, 3265920.
For there are (n+1)! permutations of decimal digits 0 through n, from which we extract n! permutations with leading zero and get n*n! = total number of integers which use each of 0-to-n decimal digits exactly once. For n=0 we have 1 integer (=0) which uses zero exactly once, hence a(0)=1 by definition.
The sequence is finite because there are only 10 decimal digits. With the initial 1 replaced by 0, we get initial terms in A001563, which is infinite. Cf. A109075 = number of primes which use each of 0-to-n decimal digits exactly once. (End)
Partial sums of A094258 -> A000142 Factorial numbers : n!=1*2*3*4*...*n 1,2,6,24,120,720,5040,40320,362880,3628800,39916800,.. [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 27 2009]
a(n) = A094304(n+1) = A01563(n-1) for n >= 2. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Oct 16 2009]
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REFERENCES
| J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
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LINKS
| Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality
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FORMULA
| a(n+1)=n*n! for n>=1.
a(n) = n!-(n-1)! for n >= 2. Also a(n) = n!-a(n-1)-a(n-2)-...-a(1) with a(1) = 1. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 14 2006
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EXAMPLE
| a(1)=1; a(2)=1*a(1)=1;
a(7)=6*(a(1)+a(2)+...+a(6))=6*(1+1+4+18+96+600)=4320.
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MAPLE
| 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 (mathar(AT)strw.leidenuniv.nl), Jul 27 2007
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MATHEMATICA
| a=s=1; lst={a}; Do[a=s*n-s; s+=a; AppendTo[lst, a], {n, 2, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 27 2009]
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CROSSREFS
| Essentially the same as A001563.
Cf. A109075.
Sequence in context: A152392 A001563 A094304 * A086681 A054139 A020072
Adjacent sequences: A094255 A094256 A094257 * A094259 A094260 A094261
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KEYWORD
| nonn
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AUTHOR
| Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 26 2004
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EXTENSIONS
| Edited by Mark Hudson, Jan 05 2005
More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 27 2007
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