%I #74 Jul 10 2023 02:29:32
%S 1,2,6,19,62,207,704,2431,8502,30056,107236,385662,1396652,5088865,
%T 18642420,68624295,253706790,941630580,3507232740,13105289370,
%U 49114150020,184560753390,695267483664,2625197720454,9933364416572,37660791173152,143048202990504
%N Second differences of Catalan numbers A000108.
%C Number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = s(2n) = 2.
%C Number of Dyck paths of semilength n+2 with no initial and no final UD's. Example: a(2)=6 because the only Dyck paths of semilength 4 with no initial and no final UD's are UUDUDUDD, UUDUUDDD, UUUDDUDD, UUUDUDDD, UUDDUUDD, UUUUDDDD. - _Emeric Deutsch_, Oct 26 2003
%C Number of branches of length 1 starting from the root in all ordered trees with n+1 edges. Example: a(1)=2 because the tree /\ has two branches of length 1 starting from the root and the path-tree of length 2 has none. a(n) = Sum_{k=0..n+1} (k*A127158(n+1,k)). - _Emeric Deutsch_, Mar 01 2007
%C Number of staircase walks from (0,0) to (n,n) that never cross y=x+2. Example: a(3) = 19 because up,up,up,right,right,right is not allowed but the other binomial(6,3)-1 = 19 paths are. - _Mark Spindler_, Nov 11 2012
%C Number of standard Young tableaux of skew shape (n+2,n)/(2), for n>=2. - _Ran Pan_, Apr 07 2015
%D S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see pp. 188, 196).
%H Vincenzo Librandi, <a href="/A026012/b026012.txt">Table of n, a(n) for n = 0..1000</a>
%H Paul Barry, <a href="https://arxiv.org/abs/2004.04577">On a Central Transform of Integer Sequences</a>, arXiv:2004.04577 [math.CO], 2020.
%H Jocelyn Quaintance and Harris Kwong, <a href="http://www.emis.de/journals/INTEGERS/papers/n29/n29.Abstract.html">A combinatorial interpretation of the Catalan and Bell number difference tables</a>, Integers, 13 (2013), #A29.
%H Murray Tannock, <a href="http://hdl.handle.net/1946/25589">Equivalence classes of mesh patterns with a dominating pattern</a>, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.
%F Expansion of (1+x^1*C^3)*C^1, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
%F a(n) = 3*(3*n^2+3*n+2)*binomial(2*n, n)/((n+1)*(n+2)*(n+3)). - _Emeric Deutsch_, Oct 26 2003
%F a(n) = Sum_{k=0..2} A039599(n,k) = A000108(n) + A000245(n) + A000344(n). - _Philippe Deléham_, Nov 12 2008
%F a(n) = binomial(2*n,n)/(n+1)*hypergeom([-2,n+1/2],[n+2],4). - _Peter Luschny_, Aug 15 2012
%F a(n) = binomial(2*n,n) - binomial(2n,n-3). - _Mark Spindler_, Nov 11 2012
%F D-finite with recurrence (n+3)*a(n) + (-5*n-6)*a(n-1) + 2*(2*n-3)*a(n-2) = 0. - _R. J. Mathar_, Jun 20 2013
%F E.g.f.: exp(2*x)*(BesselI(0,2*x) - BesselI(3,2*x)). - _Ilya Gutkovskiy_, Feb 28 2017
%F Sum_{n>=0} a(n)/4^n = 6. - _Amiram Eldar_, Jul 10 2023
%t Differences[Table[CatalanNumber[n], {n, 0, 28}], 2] (* _Jean-François Alcover_, Sep 28 2012 *)
%t Table[Binomial[2n,n]-Binomial[2n,n-3],{n,0,26}] (* _Mark Spindler_, Nov 11 2012 *)
%o (PARI) a(n) = 3*(3*n^2+3*n+2)*binomial(2*n, n)/((n+1)*(n+2)*(n+3)); /* _Joerg Arndt_, Aug 19 2012 */
%Y T(2n, n), where T is the array defined in A026009.
%Y Cf. A000108, A000245, A000344, A039599, A059346, A127158.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, _Clark Kimberling_