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%I M3015 #41 Oct 24 2023 23:15:05
%S 1,1,1,1,3,16,75,309,1183,4360,15783,56750,203929,734722,2658071,
%T 9662093,35292151,129513736,477376575,1766738922,6563071865,
%U 24464169890,91478369359,343051225066,1289887370133,4861912847046,18367285963315,69533416698304
%N a(n) = binomial(2*n-2,n-1)/n - 2^(n-1) + n.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Andrew Howroyd, <a href="/A004303/b004303.txt">Table of n, a(n) for n = 1..1000</a>
%H J. W. Moon, <a href="http://dx.doi.org/10.1016/0095-8956(76)90029-0">A problem on arcs without bypasses in tournaments,</a> J. Combinatorial Theory Ser. B 21 (1976), no. 1, 71-75. MR0427129(55 #165).
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0912.0072">Une méthode pour obtenir la fonction génératrice d'une série</a>, FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics; arXiv:0912.0072 [math.NT], 2009.
%F (n + 1)*a(n) = 68*n*a(n - 5) - 16*n*a(n - 6) + (11*n - 2)*a(n - 1) + (-47*n + 61)*a(n - 2) + (101*n - 240)*a(n - 3) + (-116*n + 398)*a(n - 4) - 304*a(n - 5) + 88*a(n - 6). - _Simon Plouffe_, Feb 09 2012
%F G.f.: x + x^2*(1 - sqrt(1-4*x) - 2*x - 2*x^3/((1-x)^2 * (1-2*x)))/(2*x^2). - _Jean-François Alcover_, Feb 13 2019
%t Table[(Binomial[2n-2,n-1])/n-2^(n-1)+n,{n,30}] (* _Harvey P. Dale_, Mar 09 2022 *)
%o (PARI) a(n) = binomial(2*n-2,n-1)/n - 2^(n-1) + n \\ _Andrew Howroyd_, Oct 24 2023
%K nonn
%O 1,5
%A _N. J. A. Sloane_
%E Extended to a(1)=1 using formula by _Alois P. Heinz_, Feb 13 2019
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