A056199 a(n) = n * a(n-1) - Sum_{k=1..n-2} a(k) with a(1) = 0 and a(2) = 1.

0, 1, 3, 11, 51, 291, 1971, 15411, 136371, 1345971, 14651571, 174318771, 2249992371, 31309422771, 467200878771, 7441464174771, 126003940206771, 2260128508782771, 42808495311726771, 853775831370606771, 17884089888607086771, 392550999147809646771

Offset: 1

Author

Robert G. Wilson v, Sep 26 1996

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..450
• R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Section B44, Springer 2010.

Crossrefs

Cf. A003422, A007489.

Programs

Magma

[0] cat [&+[Factorial(i)/3: i in [1..n]]: n in [2..25]]; // Vincenzo Librandi, Jan 17 2019

Maple

a:= proc(n) option remember; `if`(n<4, n*(n-1)/2,
      (n+1)*a(n-1) -n*a(n-2))
    end:
seq(a(n), n=1..23);  # Alois P. Heinz, Aug 11 2019

Mathematica

a[1]=0; a[2]=1; a[n_Integer] := n*a[n-1]-Sum[a[k], {k, 1, n-2}]; Table[a[n], {n, 1, 22}]
Join[{0}, Table[Plus@@(Range[n]!) / 3, {n, 2, 25}]] (* Vincenzo Librandi, Jan 17 2019 *)

Formula

a(1)=0, a(n) = (1/3)*Sum_{k=1..n} k! for n > 1. - Benoit Cloitre, Nov 12 2005

a(n) = A007489(n)/3 for n >= 2. - Philippe Deléham, Feb 10 2007

G.f.: x*(W(0)/(2-2*x)/3 -1/3), where W(k) = 1 + 1/( 1 - x*(k+2)/( x*(k+2) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 20 2013

G.f.: 1/(3*(1-x)*Q(0)) - 1/3, m=+2, where Q(k) = 1 - 2*x*(2*k+1) - m*x^2*(k+1)*(2*k+1)/( 1 - 2*x*(2*k+2) - m*x^2*(k+1)*(2*k+3)/Q(k+1) ) ; (continued fraction). - Sergei N. Gladkovskii, Sep 24 2013

Given g.f. A(x) = x^2*F(x), then F(x) = (1-x)/(1 - 4*x + 4*x^2) * (1 + x^2*F'(x)). - Paul D. Hanna, Jan 16 2019

a(n) = (n+1)*a(n-1) - n*a(n-2) for n >= 4, a(n) = n*(n-1)/2 for n < 4. - Alois P. Heinz, Aug 11 2019

Extensions

New name using a formula from Robert G. Wilson v. - Paul D. Hanna, Jan 17 2019