A054116 T(n,n-1), array T as in A054115.

1, 2, 8, 32, 152, 872, 5912, 46232, 409112, 4037912, 43954712, 522956312, 6749977112, 93928268312, 1401602636312, 22324392524312, 378011820620312, 6780385526348312, 128425485935180312, 2561327494111820312

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

For n>1, equals (-1)^(n+1) * BarnesG(n+2) times the determinant of the n X n matrix whose (i,j)-entry equals (i!-1)/i! if i=j and equals 1 otherwise. - John M. Campbell, Sep 14 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..100

Crossrefs

Equals A007489(n)-1.

Programs

Maple

a[1]:=1; a[2]:=2; for n from 3 to 20 do a[n]:=a[n-1]+factorial(n) end do; # Francesco Daddi, Aug 03 2011

Mathematica

Table[Sum[k!, {k, n}] - 1, {n, 2, 20}] (* Robert G. Wilson v, Nov 12 2004 *)

PARI

u1=1;u2=0;z=-1;for(n=3,100,u3=u2+z/n*u1;u1=u2;u2=u3;if(n>0,print1(-(u3)*n!,","))) \\ Benoit Cloitre

Formula

Let u(1)=1, u(2)=0 and u(k)=u(k-1)-1/k*u(k-2) then for n>2 a(n-1)=-u(n)*n!. - Benoit Cloitre, Nov 05 2004

a(1)=1 and, for n>=2, a(n) = sum(k=2..n, k!). - Robert G. Wilson v, Nov 12 2004

Conjecture: a(n) - (n+1)*a(n-1) + n*a(n-2) = 0. - R. J. Mathar, Jun 13 2013

G.f.: 1 - 1/(1-x) + W(0)/(1-x), where W(k) = 1 - x*(k+2)/( x*(k+2) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 25 2013

Extensions

More terms from Robert G. Wilson v, Nov 12 2004