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%I #9 Apr 25 2019 04:45:37
%S 0,1,9,116,2082,49774,1525752,58180632,2694333744,148623965136,
%T 9611353576800,719080605842400,61545135592056960,5968396255982428800,
%U 650359540100397012480,79053322881277342886400,10650510963204404347238400,1581353364394671905218406400
%N a(n) = n-th third-order hyperharmonic-exponential number, multiplied by n!.
%H Ayhan Dil and Veli Kurt, <a href="https://www.emis.de/journals/INTEGERS/papers/m38/m38.Abstract.html">Polynomials related to harmonic numbers and evaluation of harmonic number series I</a>, INTEGERS, 12 (2012), #A38.
%F a(n) = (Sum_{k=0..n} A008277(n,k) * H3(k)) * A000142(n) where H3(k) is defined by g.f.: - log(1-x)/(1-x)^3. - _Michel Marcus_, Feb 09 2013
%o (PARI)
%o hyp(n,alpha) = {x= y+O(y^(n+1)); gf = - log(1-x)/(1-x)^alpha; polcoeff(gf, n, y);}
%o a(n, alpha=3) = sum(k=0, n, n!*(sum(i=0, k, (-1)^i*binomial(k, i)*i^n)*(-1)^k/k!)*hyp(k,alpha));
%o \\ _Michel Marcus_, Feb 09 2013
%K nonn
%O 0,3
%A _Michel Marcus_, following a suggestion of _N. J. A. Sloane_, Feb 09 2013
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