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 A014288 [ Sum k!/2, k=0..n ], or floor( A003422(n+1)/2 ). 9
 0, 1, 2, 5, 17, 77, 437, 2957, 23117, 204557, 2018957, 21977357, 261478157, 3374988557, 46964134157, 700801318157, 11162196262157, 189005910310157, 3390192763174157, 64212742967590157, 1280663747055910157, 26826134832910630157, 588826498721714470157 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The first term a(0) would be a fraction if the floor( ... ) function would be omitted ; for n>=2, all terms from A003422 are even. - M. F. Hasler, Dec 16 2007 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..200 FORMULA 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 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] 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 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 a(n) ~ n!/2. - Vaclav Kotesovec, Aug 10 2013 E.g.f.: -1/2 + exp(x)/2*Sum(k>=0, k! - k*Gamma(k,x)). - Robert Israel, Jun 01 2015 MAPLE a:= proc(n) a(n):= `if`(n<3, n, (n+1)*a(n-1)-n*a(n-2)) end: seq(a(n), n=0..25);  # Alois P. Heinz, Feb 01 2013 MATHEMATICA f[x_] := {Floor[1 + (n - x[[2]])*x[[1]]], x[[2]] + 1}; 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 *) 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 *) PROG (PARI) A014288(n)=sum(k=0, n, k!)>>1 \\ M. F. Hasler, Dec 16 2007 CROSSREFS Cf. A003422, A067078, A007489. Sequence in context: A118100 A129591 A099825 * A199164 A184509 A020096 Adjacent sequences:  A014285 A014286 A014287 * A014289 A014290 A014291 KEYWORD nonn AUTHOR EXTENSIONS Edited by M. F. Hasler, Dec 16 2007 STATUS approved

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