A067078 a(1) = 1, a(2) = 2, a(n) = (n-1)*a(n-1) - (n-2)*a(n-2).

1, 2, 3, 5, 11, 35, 155, 875, 5915, 46235, 409115, 4037915, 43954715, 522956315, 6749977115, 93928268315, 1401602636315, 22324392524315, 378011820620315, 6780385526348315, 128425485935180315, 2561327494111820315

Offset: 1

Author

Amarnath Murthy, Jan 05 2002

Keywords

• nonn

Comments

Successive differences are factorials, or (n+1)st successive difference divided by n-th successive difference = n. I.e., {a(n+2)-a(n+1)}/{a(n+1)-a(n)} = n. - Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 14 2003

Equals the row sums of A165680. - Johannes W. Meijer, Oct 16 2009

Examples

			a(6) = 35, a(5)= 11 hence a(7) = 6*35 - 5*11 = 155.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A003422, A014288, A007489, A165680.

Programs

Haskell

a067078 n = a067078_list !! (n-1)
a067078_list = scanl (+) 1 a000142_list
-- Reinhard Zumkeller, Dec 27 2011

Mathematica

a[1] = 1; a[2] = 2; a[n_] := a[n] = (n - 1)*a[n - 1] - (n - 2)*a[n - 2]; Table[ a[n], {n, 1, 25} ]
a=FoldList[Plus,2,(Range@40)! ];PrependTo[a,1] (* Vladimir Joseph Stephan Orlovsky, May 21 2010 *)

PARI

A067078(n)=sum(k=0, n-2, k!, 1) \\ M. F. Hasler, Dec 16 2007

Formula

a(n) = 1 + Sum_{i=0..n-2} i! = 2*A014288(n-1)+1 = A007489(n-2)+2 (n>1). - Henry Bottomley, Oct 23 2002; corrected by M. F. Hasler, Dec 16 2007

a(n) = 1+!(n-1) = 1+A003422(n-1); a(n+1)=a(n)+(n-1)!. - M. F. Hasler, Dec 16 2007

E.g.f.: A(x)=x*B(x) satisfies the differential equation B'(x)=B(x)+log(1/(1-x))+1. - Vladimir Kruchinin, Jan 19 2011

Extensions

More terms from Robert G. Wilson v, Jan 07 2002

Edited by M. F. Hasler, Dec 16 2007