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

1,2

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

Links

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

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

Example

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

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 *)

Prog

(PARI) A067078(n)=sum(k=0, n-2, k!, 1) \\ M. F. Hasler, Dec 16 2007
(Haskell)
a067078 n = a067078_list !! (n-1)
a067078_list = scanl (+) 1 a000142_list
-- Reinhard Zumkeller, Dec 27 2011

Crossrefs

Cf. A003422, A014288, A007489, A165680.

Sequence in context: A305971 A064095 A061935 * A124561 A355503 A167604

Adjacent sequences: A067075 A067076 A067077 * A067079 A067080 A067081

Keyword

nonn

Author

Amarnath Murthy, Jan 05 2002

Extensions

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

Edited by M. F. Hasler, Dec 16 2007

Status

approved