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%I #20 Sep 26 2015 13:56:59
%S 1,1,5,24,134,866,6392,53198,493628,5057522,56741240,692118422,
%T 9122245508,129220379978,1958059133552,31607140330670,541515698082332,
%U 9814691158604258,187629572002767848,3773371262361852422,79636835475910932020
%N a(n) = Sum_{i=0..n-1} (n+i)*a(n-1-i) for n>1, a(0)=1, a(1)=1.
%C Occurs in making the Q-residue A193657.
%C Second difference of A002627. - _Peter Luschny_, May 30 2014
%H Vincenzo Librandi, <a href="/A193668/b193668.txt">Table of n, a(n) for n = 0..200</a>
%F Recurrence: a(n) = (n+2)*a(n-1) - (n-2)*a(n-2). - _Vaclav Kotesovec_, Nov 20 2012
%F a(n) ~ n!*n*(e-1). - _Vaclav Kotesovec_, Nov 20 2012
%F a(n) = (n-n^2-1)*Gamma(n) + e*(n*Gamma(n+1,1)-(n-1)*Gamma(n,1)) for n>0. - _Peter Luschny_, May 30 2014.
%p a := n -> `if`(n=0,1,(n-n^2-1)*GAMMA(n)+exp(1)*((1-n)*GAMMA(n,1) + n*GAMMA(n+1, 1))): seq(simplify(a(n)),n=0..20); # _Peter Luschny_, May 30 2014
%t (See A193657.)
%t Flatten[{1,RecurrenceTable[{(n-2)*a[n-2] - (n+2)*a[n-1] + a[n] == 0, a[1]==1, a[2]==5}, a, {n, 20}]}] (* _Vaclav Kotesovec_, Nov 20 2012 *)
%t CoefficientList[Series[Log[x-1]+E*Gamma[0,1-x]-E*Gamma[0,1]+1-I*Pi+(E^x*x-x^2)/(x-1)^2, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Nov 20 2012 *)
%o (PARI) a(n)=if(n<2,1,sum(i=0,n-1,(n+i)*a(n-1-i))) \\ _Charles R Greathouse IV_, May 30 2014
%Y Cf. A193657, A002627.
%K nonn
%O 0,3
%A _Clark Kimberling_, Aug 02 2011
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