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%I #62 Nov 02 2023 14:20:16
%S 0,1,2,5,17,77,437,2957,23117,204557,2018957,21977357,261478157,
%T 3374988557,46964134157,700801318157,11162196262157,189005910310157,
%U 3390192763174157,64212742967590157,1280663747055910157,26826134832910630157,588826498721714470157
%N a(n) = floor(Sum_{k=0..n} k!/2), or floor( A003422(n+1)/2 ).
%C The first term a(0) would be a fraction if the floor( ... ) function were omitted; for n >= 2, all terms from A003422 are even. - _M. F. Hasler_, Dec 16 2007
%H Alois P. Heinz, <a href="/A014288/b014288.txt">Table of n, a(n) for n = 0..200</a>
%F a(0)=0, a(1)=1, a(2)=2, a(n) = (n+1)*a(n-1) - n*a(n-2). - _Benoit Cloitre_, Sep 07 2002
%F a(0) = 0, a(n) = (1/2)*floor(1 + 1*floor(1 + 2*floor(1 + ... + (n-1)*floor(1+n*floor(1))). - Joseph E. Cooper III (easonrevant(AT)gmail.com), Aug 19 2008
%F G.f.: G(0)/(1-x)/2 -1/2, where G(k)= 1 + (2*k + 1)*x/( 1 - 2*x*(k+1)/(2*x*(k+1) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 24 2013
%F G.f.: A(x) = (Sum_{n>=0} x^n*n!)/(2-2*x) - 1/2 = G(0)/(4*(1-x)) - 1/2, where G(k) = 1 + 1/(1 - x/(x + 1/(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 02 2013
%F a(n) ~ n!/2. - _Vaclav Kotesovec_, Aug 10 2013
%F E.g.f.: -1/2 + (exp(x)/2)*Sum_{k>=0} (k! - k*Gamma(k,x)). - _Robert Israel_, Jun 01 2015
%F a(n) = ((n+1)!*ExpIntegral(n+2,-1)+Ei(1)+Pi*i)/(2*e). - _Ammar Khatab_, Aug 14 2020
%p a:= proc(n) a(n):= `if`(n<3, n, (n+1)*a(n-1)-n*a(n-2)) end:
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Feb 01 2013
%t f[x_] := {Floor[1 + (n - x[[2]])*x[[1]]], x[[2]] + 1};
%t a[0] = 0; a[n_] := Nest[f, {1, 0}, n][[1]]/2 (* Joseph E. Cooper III (easonrevant(AT)gmail.com), Aug 19 2008 *) (* updated by _Jean-François Alcover_, Jun 01 2015 *)
%t a[n_]:=-(1/2) Subfactorial[-1]-1/2(-1)^n Gamma[2+n] Subfactorial[-2-n]; Table[a[n] //FullSimplify,{n,0,25}] (* _Gerry Martens_, May 29 2015 *)
%o (PARI) A014288(n)=sum(k=0,n,k!)>>1 \\ _M. F. Hasler_, Dec 16 2007
%o (Magma) [Floor((&+[Factorial(j): j in [0..n]])/2): n in [0..30]]; // _G. C. Greubel_, Sep 05 2022
%o (SageMath) [sum(factorial(j) for j in (0..n))//2 for n in (0..30)] # _G. C. Greubel_, Sep 05 2022
%o (Python)
%o from math import factorial
%o def A014288(n): return sum(factorial(k) for k in range(n+1))>>1 # _Chai Wah Wu_, Nov 01 2023
%Y Cf. A003422, A007489, A067078.
%K nonn
%O 0,3
%A _N. J. A. Sloane_
%E Edited by _M. F. Hasler_, Dec 16 2007
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