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%I #61 Jan 15 2024 01:44:00
%S 1,2,7,26,101,406,1676,7066,30302,131782,579867,2576982,11550237,
%T 52152330,237005385,1083211410,4975796735,22960105510,106377393365,
%U 494674698190,2308015808015,10801388134690,50691017885290,238503869991926,1124828963516896,5316520644648026,25179670936870021
%N Expansion of g.f.: ((1 - x - sqrt(1-6*x+5*x^2))/(2*x))^2.
%C Convolution of A002212 with itself.
%C Number of skew Dyck paths of semilength n+1 starting with UU. A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. Example: a(2)=7 because we have UUDDUD, UUDUDD, UUDUDL, UUUDDD, UUUDDL, UUUDLD and UUUDLL. - _Emeric Deutsch_, May 11 2007
%C a(n) is also the number of path-pairs (u,v) having the following six properties: 1) the lengths of u and v sum up to 2n, 2) u and v both start at (0,0), 3) (0,0) is the only vertex that u and v have in common, 4) the steps that u can make are (1,0), (0,1) and (0,-1), 5) the steps that v can make are (1,0), (-1,0) and (0,1), 6) if A and B are the termini of u and v, respectively, then B=A+(1,-1). - _Svjetlan Feretic_, Jun 09 2013
%H Vincenzo Librandi, <a href="/A045868/b045868.txt">Table of n, a(n) for n = 0..1000</a>
%H S. J. Cyvin et al., <a href="http://dx.doi.org/10.1021/ci00021a026">Enumeration and classification of certain polygonal systems representing polycyclic conjugated hydrocarbons: annelated catafusenes</a>, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.
%H E. Deutsch, E. Munarini, and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.015">Skew Dyck paths</a>, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
%F a(n) = (2/n)*Sum_{j=1..n} binomial(n, j)*binomial(2j+1, j-1) for n >= 1.
%F a(n) = A055450(n, n-1).
%F D-finite with recurrence: (n+2)*a(n) = (6*n+2)*a(n-1) - (5*n-10)*a(n-2). - _Vladeta Jovovic_, Jul 16 2004
%F a(n) = 2*Hypergeometric2F1(5/2, 1-n, 4, -4). - _Jean-François Alcover_, Apr 30 2012
%F a(n) ~ 2*5^(n+1/2)/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 08 2012
%F G.f.: 1 - 1/x + Q(0)*(1-x)/x, where Q(k) = 1 + (4*k+1)*x/((1-x)*(k+1) - x*(1-x)*(2*k+2)*(4*k+3)/(x*(8*k+6)+(2*k+3)*(1-x)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 14 2013
%F G.f.: 1/x - 1 - 2*(1-x)/x/( G(0) + 1), where G(k) = 1 + 2*x*(4*k+1)/( (2*k+1)*(1-x) - x*(1-x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1-x)*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 24 2013
%p a := n->(2/n)*sum(binomial(n,j)*binomial(2*j+1,j-1),j=1..n): 1,seq(a(n),n=1..22);
%t a[n_] := 2*Hypergeometric2F1[ 5/2, 1-n, 4, -4]; a[0] = 1; Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Apr 30 2012, after Maple *)
%o (PARI) a(n)=polcoeff((1-x-sqrt(1-6*x+5*x^2+x^2*O(x^n)))^2/4,n+2)
%o (PARI) my(x='x+O('x^66)); Vec(((1-x-sqrt(1-6*x+5*x^2))/(2*x))^2) \\ _Joerg Arndt_, May 04 2013
%o (Magma) [n le 2 select n else (2*(3*n-2)*Self(n-1) - 5*(n-3)*Self(n-2))/(n+1): n in [1..30]]; // _G. C. Greubel_, Jan 12 2024
%o (SageMath)
%o def A045868(n): return 1 if n==0 else (2/n)*sum( binomial(n,j)*binomial(2*j+1,j-1) for j in range(1,n+1))
%o [A045868(n) for n in range(31)] # _G. C. Greubel_, Jan 12 2024
%Y Cf. A055450.
%Y Essentially the first differences of A002212 and A025238.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Emeric Deutsch_, May 11 2007
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