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A094258
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a(1) = 1, a(n+1) = n*n! for n >= 1.
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9
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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
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OFFSET
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1,3
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COMMENTS
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The old definition was: "a(1) = 1, a(n+1) = n*(a(1) + a(2) + ... + a(n))."
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
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
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.
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)
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LINKS
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FORMULA
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a(n+1) = n*n! = A001563(n) for n>=1.
a(n) = n! - (n-1)! for n >= 2.
a(n) = n! - a(n-1) - a(n-2) - ... - a(1). with a(1) = 1. (End)
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
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
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EXAMPLE
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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
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MATHEMATICA
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Module[{lst={1}}, Do[AppendTo[lst, n*Total[lst]], {n, 30}]; lst] (* Harvey P. Dale, Jul 01 2012 *)
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PROG
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CROSSREFS
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Up to the offset and initial value, the same as A001563, cf. formula.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Edited by Mark Hudson, Jan 05 2005
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STATUS
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approved
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