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A093344
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a(n) = n! * Sum_{i=1..n} (1/i)*Sum_{j=0..i-1} 1/j!.
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6
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0, 1, 4, 17, 84, 485, 3236, 24609, 210572, 2004749, 21033900, 241237001, 3003349124, 40345599957, 581765196884, 8963453118065, 146969877361116, 2555361954692189, 46963373856864092, 909707559383702169, 18524816853636447380, 395634467245613474981
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = Integral_{x=1..oo} exp(1-x)*(x^n*log(x) - n!/x) dx. - Groux Roland, Mar 12 2011
a(n) = exp(1)*(H(n)*n! + (Ei(-1)-gamma)*n! + hypergeom([n+1,n+1],[n+2,n+2],-1)/(n+1)^2), where H(n)*n! = A000254(n), -Ei(-1) is A099285, gamma is A001620.
Recurrence: a(0) = 0, a(1) = 1, a(2) = 4, a(n) = 2*n*a(n-1) + (2-n^2)*a(n-2) + (n-2)^2*a(n-3).
(End)
a(n) = n!*e*Sum_{k=1..n} Gamma(k,1)/k!. - Robert Israel, Oct 28 2015
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MAPLE
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f:= gfun:-rectoproc({a(0) = 0, a(1) = 1, a(2) = 4, a(n) = 2*n*a(n-1) + (2-n^2)*a(n-2) + (n-2)^2*a(n-3)}, a(n), remember):
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MATHEMATICA
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Round@Table[E n! Sum[Gamma[k, 1]/k!, {k, 1, n}], {n, 0, 20}]
Round@Table[E ((HarmonicNumber[n] + ExpIntegralEi[-1] - EulerGamma) n! + HypergeometricPFQ[{n+1, n+1}, {n+2, n+2}, -1]/(n+1)^2), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 28 2015 *)
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PROG
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(PARI) a(n) = n!*sum(i=1, n, 1/i*sum(j=0, i-1, 1/j!))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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