

A045868


Expansion of g.f.: ((1  x  sqrt(16*x+5*x^2))/(2*x))^2.


5



1, 2, 7, 26, 101, 406, 1676, 7066, 30302, 131782, 579867, 2576982, 11550237, 52152330, 237005385, 1083211410, 4975796735, 22960105510, 106377393365, 494674698190, 2308015808015, 10801388134690, 50691017885290, 238503869991926, 1124828963516896, 5316520644648026, 25179670936870021
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OFFSET

0,2


COMMENTS

Convolution of A002212 with itself.
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 xaxis, 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
a(n) is also the number of pathpairs (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


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. J. Cyvin et al., Enumeration and classification of certain polygonal systems representing polycyclic conjugated hydrocarbons: annelated catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 11741180.
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 21912203.


FORMULA

a(n) = (2/n)*Sum_{j=1..n} binomial(n, j)*binomial(2j+1, j1) for n >= 1.
Dfinite with recurrence: (n+2)*a(n) = (6*n+2)*a(n1)  (5*n10)*a(n2).  Vladeta Jovovic, Jul 16 2004
a(n) ~ 2*5^(n+1/2)/(sqrt(Pi)*n^(3/2)).  Vaclav Kotesovec, Oct 08 2012
G.f.: 1  1/x + Q(0)*(1x)/x, where Q(k) = 1 + (4*k+1)*x/((1x)*(k+1)  x*(1x)*(2*k+2)*(4*k+3)/(x*(8*k+6)+(2*k+3)*(1x)/Q(k+1))); (continued fraction).  Sergei N. Gladkovskii, May 14 2013
G.f.: 1/x  1  2*(1x)/x/( G(0) + 1), where G(k) = 1 + 2*x*(4*k+1)/( (2*k+1)*(1x)  x*(1x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1x)*(k+1)/G(k+1))); (continued fraction).  Sergei N. Gladkovskii, Jun 24 2013


MAPLE

a := n>(2/n)*sum(binomial(n, j)*binomial(2*j+1, j1), j=1..n): 1, seq(a(n), n=1..22);


MATHEMATICA

a[n_] := 2*Hypergeometric2F1[ 5/2, 1n, 4, 4]; a[0] = 1; Table[a[n], {n, 0, 22}] (* JeanFrançois Alcover, Apr 30 2012, after Maple *)


PROG

(PARI) a(n)=polcoeff((1xsqrt(16*x+5*x^2+x^2*O(x^n)))^2/4, n+2)
(PARI) x='x+O('x^66); Vec(((1xsqrt(16*x+5*x^2))/(2*x))^2) \\ Joerg Arndt, May 04 2013


CROSSREFS

T(n, n1) where T is A055450.
Essentially the first differences of A002212 and A025238.
Sequence in context: A279002 A176280 A349713 * A171711 A129482 A150536
Adjacent sequences: A045865 A045866 A045867 * A045869 A045870 A045871


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Emeric Deutsch, May 11 2007


STATUS

approved



