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A014144
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Apply partial sum operator twice to factorials.
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8
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0, 1, 3, 7, 17, 51, 205, 1079, 6993, 53227, 462341, 4500255, 48454969, 571411283, 7321388397, 101249656711, 1502852293025, 23827244817339, 401839065437653, 7182224591785967, 135607710526966281, 2696935204638786595, 56349204870460046909, 1234002202313888987223
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OFFSET
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0,3
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COMMENTS
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If s(n) is a sequence defined as s(0)=a, s(1)=b, s(n) = n*(s(n-1) - s(n-2)), n>1, then s(n) = n*b - (a(n)-1)*a. - Gary Detlefs, Feb 23 2011
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LINKS
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FORMULA
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a(n) = (n-1) * !n - n! + 1, !n = Sum_{k=0..n-1} k!. - Joe Keane (jgk(AT)jgk.org)
G.f.: x*G(0)/(1-x)^2, where G(k)= 1 + (2*k + 1)*x/( 1 - 2*x*(k+1)/(2*x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
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MAPLE
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b:= proc(n) option remember; `if`(n<0, [0$2],
(q-> (f-> [f[2]+q, q]+f)(b(n-1)))(n!))
end:
a:= n-> b(n-1)[1]:
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MATHEMATICA
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PROG
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(Magma) [(k-1)*(&+[Factorial(j): j in [0..k-1]]) - Factorial(k) + 1: k in [1..25]]; // G. C. Greubel, Sep 03 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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