A004303 a(n) = binomial(2*n-2,n-1)/n - 2^(n-1) + n. (Formerly M3015)

1, 1, 1, 1, 3, 16, 75, 309, 1183, 4360, 15783, 56750, 203929, 734722, 2658071, 9662093, 35292151, 129513736, 477376575, 1766738922, 6563071865, 24464169890, 91478369359, 343051225066, 1289887370133, 4861912847046, 18367285963315, 69533416698304

Offset

1,5

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

J. W. Moon, A problem on arcs without bypasses in tournaments, J. Combinatorial Theory Ser. B 21 (1976), no. 1, 71-75. MR0427129(55 #165).

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série, FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics; arXiv:0912.0072 [math.NT], 2009.

Formula

(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

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

Mathematica

Table[(Binomial[2n-2, n-1])/n-2^(n-1)+n, {n, 30}] (* Harvey P. Dale, Mar 09 2022 *)

Prog

(PARI) a(n) = binomial(2*n-2, n-1)/n - 2^(n-1) + n \\ Andrew Howroyd, Oct 24 2023

Crossrefs

Sequence in context: A316170 A038602 A221829 * A335625 A317365 A207836

Adjacent sequences: A004300 A004301 A004302 * A004304 A004305 A004306

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Extended to a(1)=1 using formula by Alois P. Heinz, Feb 13 2019

Status

approved