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%I #48 Jan 18 2023 18:50:18
%S 0,1,3,11,51,291,1971,15411,136371,1345971,14651571,174318771,
%T 2249992371,31309422771,467200878771,7441464174771,126003940206771,
%U 2260128508782771,42808495311726771,853775831370606771,17884089888607086771,392550999147809646771
%N a(n) = n * a(n-1) - Sum_{k=1..n-2} a(k) with a(1) = 0 and a(2) = 1.
%H Vincenzo Librandi, <a href="/A056199/b056199.txt">Table of n, a(n) for n = 1..450</a>
%H R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Section B44, Springer 2010.
%F a(1)=0, a(n) = (1/3)*Sum_{k=1..n} k! for n > 1. - _Benoit Cloitre_, Nov 12 2005
%F a(n) = A007489(n)/3 for n >= 2. - _Philippe Deléham_, Feb 10 2007
%F 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
%F 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
%F 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
%F 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
%p a:= proc(n) option remember; `if`(n<4, n*(n-1)/2,
%p (n+1)*a(n-1) -n*a(n-2))
%p end:
%p seq(a(n), n=1..23); # _Alois P. Heinz_, Aug 11 2019
%t 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}]
%t Join[{0}, Table[Plus@@(Range[n]!) / 3, {n, 2, 25}]] (* _Vincenzo Librandi_, Jan 17 2019 *)
%o (Magma) [0] cat [&+[Factorial(i)/3: i in [1..n]]: n in [2..25]]; // _Vincenzo Librandi_, Jan 17 2019
%Y Cf. A003422, A007489.
%K easy,nonn
%O 1,3
%A _Robert G. Wilson v_, Sep 26 1996
%E New name using a formula from _Robert G. Wilson v_. - _Paul D. Hanna_, Jan 17 2019
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