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 A065097 a(n) = ((2n+1)+(2n-1)-1)!/((2n+1)!*(2n-1)!). 12

%I

%S 1,7,66,715,8398,104006,1337220,17678835,238819350,3282060210,

%T 45741281820,644952073662,9183676536076,131873975875180,

%U 1907493251046152,27767032438524099,406472021074865382,5979899192930226746,88366931393503350700,1311063521138246054410

%N a(n) = ((2n+1)+(2n-1)-1)!/((2n+1)!*(2n-1)!).

%C A Catalan-like formula using consecutive odd numbers. Recall that Catalan numbers (A000108) are given by ((n+1)+(n)-1)!/((n+1)!(n)!).

%C From _David Callan_, Jun 01 2006: (Start)

%C a(n) = number of Dyck (2n)-paths (i.e., semilength = 2n) all of whose interior returns to ground level (if any) occur at or before the (2n-2)-nd step, that is, they occur strictly before the midpoint of the path.

%C For example, a(2)=7 counts UUUUDDDD, UUUDUDDD, UUDUUDDD, UUDUDUDD, UUUDDUDD, UD.UUUDDD, UD.UUDUDD ("." denotes an interior return to ground level).

%C This result follows immediately from an involution on Dyck paths, due to Emeric Deutsch, defined by E->E, UPDQ -> UQDP (where E is the empty Dyck path; U=upstep, D=downstep and P,Q are arbitrary Dyck paths), because the involution is fixed-point-free on Dyck (2n)-paths and contains one path of the type being counted in each orbit.

%C a(n) = Sum_{k=0..n-1}C(2n-1-2k)C(2k). This identity has the following combinatorial interpretation:

%C a(n) = number of odd-GL-marked Dyck (2n-1)-paths. An odd-GL vertex is a vertex at location (2i,0) for some odd i>=1 (path starts at origin). An odd-GL-marked Dyck path is a Dyck path with one of its odd-GL vertices marked. For example, a(2)=7 counts UUUDDD*, UUDUDD*, UD*UUDD, UDUUDD*, UD*UDUD, UDUDUD*, UUDDUD* (the * denotes the marked odd-GL vertex). (End)

%C a(n+1) = Sum_{k=0..n}C(k)*C(2*n+1-k), n>=0, with C(n) = A000108(n), gives also the odd part of the bisection of the half-convolution of the Catalan sequence A000108 with itself. For the definition of the half-convolution of a sequence with itself see a comment on A201204. There one also finds the rule for the o.g.f. given below in the formula section. The even part of this bisection is found under A201205. - _Wolfdieter Lang_, Jan 05 2012

%C From _Peter Bala_, Dec 01 2015: (Start)

%C Let x = p/q be a positive rational in reduced form with p,q > 0. Define Cat(x) = 1/(2*p + q)*binomial(2*p + q, p). Then Cat(n) = Catalan(n). This sequence is Cat(n + 1/2) = 1/(4*n + 4)*binomial(4*n + 4, 2*n + 1). Cf. A265101 (Cat(n + 1/3), A265102 (Cat(n + 1/4)) and A265103 (Cat(n + 1/4)).

%C Number of maximal faces of the rational associahedron Ass(2*n + 1, 2*n + 3). Number of lattice paths from (0, 0) to (2*n + 3, 2*n + 1) using steps of the form (1, 0) and (0, 1) and staying above the line y = (2*n + 1)/(2*n + 3)*x. See Armstrong et al. (End)

%H Harry J. Smith, <a href="/A065097/b065097.txt">Table of n, a(n) for n=1..100</a>

%H D. Armstrong, B. Rhoades, and N. Williams, <a href="http://arxiv.org/abs/1305.7286">Rational associahedra and noncrossing partitions</a> arxiv:1305.7286v1 [math.CO], 2008.

%H E. Deutsch, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00370-7">An involution on Dyck paths and its consequences</a>, Discrete Math., 204 (1999), no. 1-3, 163-166.

%F a(n) = binomial(4*n-1, 2*n-1)/(2*n+1).

%F a(n) = C(2n)/2 where C(n) is the Catalan number A000108. - _David Callan_, Jun 01 2006

%F G.f.: (sqrt(2)/2)/sqrt(1+sqrt(1-16*x))-1/2. - _Vladeta Jovovic_, Sep 26 2003

%F G.f.: 3F2( (1, 5/4, 7/4); (2, 5/2) )(16*x). - _Olivier GĂ©rard_, Feb 16 2011

%F O.g.f.: ((cata(sqrt(x)) + cata(-sqrt(x)))/2 -1)/2, with the o.g.f. cata(x) of the Catalan numbers. See the W. Lang comment above. - _Wolfdieter Lang_, Jan 05 2012

%F a(n) = hypergeometric([1-2*n,-2*n],[2],1)/2. - _Peter Luschny_, Sep 22 2014

%F a(n) = A001448(n) / (4*n + 2) if n>0. - _Michael Somos_, Oct 25 2014

%F n*(2*n+1)*a(n) -2*(4*n-1)*(4*n-3)*a(n-1)=0. - _R. J. Mathar_, Oct 31 2015

%F O.g.f. is Revert( x*(1 + x)/(1 + 2*x)^4 ). - _Peter Bala_, Dec 01 2015

%e G.f.: x + 7*x^2 + 66*x^3 + 715*x^4 + 8398*x^5 + 104006*x^6 + ...

%p seq(binomial(4*n-1,2*n-1)/(2*n+1), n=1..30); # _Robert Israel_, Dec 08 2015

%t a[ n_] := If[ n < 1, 0, Binomial[ 4 n - 1, 2 n - 1] / (2 n + 1)]; (* _Michael Somos_, Oct 25 2014 *)

%o (MuPAD) combinat::dyckWords::count(2*n)/2 \$ n = 1..26 // _Zerinvary Lajos_, Apr 25 2007

%o (PARI) { for (n=1, 100, a=(4*n - 1)!/((2*n + 1)!*(2*n - 1)!); write("b065097.txt", n, " ", a) ) } \\ _Harry J. Smith_, Oct 07 2009

%o (PARI) vector(100, n, binomial(4*n-1, 2*n-1)/(2*n+1)) \\ _Altug Alkan_, Dec 08 2015

%o (Sage)

%o A065097 = lambda n: hypergeometric([1-2*n,-2*n],[2],1)/2

%o [Integer(A065097(n).n(500)) for n in (1..20)] # _Peter Luschny_, Sep 22 2014

%o (MAGMA) [Binomial(4*n-1, 2*n-1)/(2*n+1): n in [1..20]]; // _Vincenzo Librandi_, Dec 09 2015

%Y Cf. A003150 (for analog with consecutive Fibonacci numbers).

%Y Equals (1/2) A048990. Cf. A001795, A201204, A201205, A265101, A265102, A265103.

%K nonn,easy

%O 1,2

%A _Len Smiley_, Nov 11 2001

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Last modified September 26 02:28 EDT 2020. Contains 337346 sequences. (Running on oeis4.)