A193668 a(n) = Sum_{i=0..n-1} (n+i)*a(n-1-i) for n>1, a(0)=1, a(1)=1.

1, 1, 5, 24, 134, 866, 6392, 53198, 493628, 5057522, 56741240, 692118422, 9122245508, 129220379978, 1958059133552, 31607140330670, 541515698082332, 9814691158604258, 187629572002767848, 3773371262361852422, 79636835475910932020

Offset: 0

Author

Clark Kimberling, Aug 02 2011

Keywords

• nonn

Comments

Occurs in making the Q-residue A193657.

Second difference of A002627. - Peter Luschny, May 30 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A193657, A002627.

Programs

Maple

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

Mathematica

(See A193657.)
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 *)
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 *)

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

Formula

Recurrence: a(n) = (n+2)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Nov 20 2012

a(n) ~ n!*n*(e-1). - Vaclav Kotesovec, Nov 20 2012

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.