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a(n) = Sum_{k=0..floor((n-1)/2)} T(n,k) * T(n,k+1), with T given by A008315.
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%I #42 Nov 22 2024 07:34:02

%S 1,2,9,24,95,286,1099,3536,13479,45220,172150,594320,2265003,7983990,

%T 30487175,109174560,417812417,1514797020,5810065898,21275014800,

%U 81775140083,301892460012,1162703549474,4321730134624,16675372590850,62340424959176,240949471232124

%N a(n) = Sum_{k=0..floor((n-1)/2)} T(n,k) * T(n,k+1), with T given by A008315.

%C a(n) is the number of Dyck (n+2)-paths with UU spanning the midpoint. E.g., for n=2 the two Dyck 4-paths are UUDU.UDDD and UDUU.UDDD where dot marks the midpoint. - _David Scambler_, Feb 11 2011

%C Apparently also the number of returns to the left of or to the midpoint of all Dyck paths with semilength n+1. - _David Scambler_, Apr 30 2013

%H Alon Regev, <a href="https://arxiv.org/abs/1210.3349">The central component of a triangulation</a>, arXiv:1210.3349 [math.CO], 2012, see p. 6.

%H Alon Regev, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Regev/regev4.html">The Central Component of a Triangulation</a>, J. Int. Seq. 16 (2013) #13.4.1

%F Conjecture D-finite with recurrence -(n+2)*(13*n-2)*(3+n)^2*a(n) +10*(8*n^2+3*n-8)*(n+2)^2*a(n-1) +8*(12*n^4+47*n^3+52*n^2+67*n+20)*a(n-2) -160*(8*n^2+3*n-8)*(n-1)^2*a(n-3) +128*(7*n+4)*(2*n-5)*(-2+n)^2*a(n-4)=0. - _R. J. Mathar_, Nov 22 2024

%t a[n_] := With[{C = CatalanNumber}, Sum[C[k]*C[n+1-k], {k, 1, (n+1)/2}]]; Array[a, 30] (* _Jean-François Alcover_, May 01 2017 *)

%o (Sage)

%o def C(n): return binomial(2*n,n)/(n+1) # Catalan numbers

%o def A027302(n): return add(C(k)*C(n+1-k) for k in (1..(n+1)/2))

%o [A027302(n) for n in (1..22)] # _Peter Luschny_, Jun 27 2013

%K nonn

%O 1,2

%A _Clark Kimberling_

%E More terms from _Sean A. Irvine_, Oct 26 2019