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A196532 a(n) = (n+1)!*(H(n)+H(n+1)), where H(n) = Sum_{k=1..n} 1/k is the n-th harmonic number.

%I A196532 #17 Mar 28 2018 21:59:15
%S A196532 1,5,20,94,524,3408,25416,214128,2012832,20894400,237458880,
%T A196532 2932968960,39126516480,560704273920,8591147712000,140160890419200,
%U A196532 2425888391270400,44398288688947200,856727919929548800
%N A196532 a(n) = (n+1)!*(H(n)+H(n+1)), where H(n) = Sum_{k=1..n} 1/k is the n-th harmonic number.
%C A196532 Denominator of a(n)/n! is listed in A096620.
%C A196532 a(n) - (n+1)*a(n-1) = A129326(n), n > 0. - _Gary Detlefs_, Oct 04 2011
%H A196532 Robert Israel, <a href="/A196532/b196532.txt">Table of n, a(n) for n = 0..448</a>
%F A196532 From _Robert Israel_, Mar 28 2018: (Start)
%F A196532 E.g.f.: (1+x - 2*log(1-x))/(1-x)^2.
%F A196532 a(n+3) = (3*n+8)*a(n+2) - (3*n+7)*(n+2)*a(n+1) + (n+1)*(n+2)^2*a(n). (End)
%p A196532 H:= n-> sum(1/k,k=1..n):seq((n+1)!*(H(n+1)+H(n)), n=0..20);
%p A196532 # Alternative:
%p A196532 f:= gfun:-rectoproc({a(n+3) = (3*n+8)*a(n+2)-(3*n+7)*(n+2)*a(n+1)+(n+1)*(n+2)^2*a(n),a(0)=1,a(1)=5,a(2)=20},a(n),remember):
%p A196532 map(f, [$0..50]); # _Robert Israel_, Mar 28 2018
%t A196532 Table[(n+1)!Total[HarmonicNumber[{n,n+1}]],{n,0,20}] (* _Harvey P. Dale_, Jul 17 2013 *)
%K A196532 nonn
%O A196532 0,2
%A A196532 _Gary Detlefs_, Oct 03 2011