A045868 Expansion of g.f.: ((1 - x - sqrt(1-6*x+5*x^2))/(2*x))^2.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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 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

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

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), 1174-1180.
• Emeric Deutsch, Emanuele Munarini, and Simone Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
• Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 99.

Crossrefs

Cf. A055450.

Essentially the first differences of A002212 and A025238.

Programs

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

Maple

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

Mathematica

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 *)

PARI

a(n)=polcoeff((1-x-sqrt(1-6*x+5*x^2+x^2*O(x^n)))^2/4,n+2)

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

SageMath

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))
[A045868(n) for n in range(31)] # G. C. Greubel, Jan 12 2024

Formula

a(n) = (2/n)*Sum_{j=1..n} binomial(n, j)*binomial(2j+1, j-1) for n >= 1.

a(n) = A055450(n, n-1).

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

a(n) = 2*Hypergeometric2F1(5/2, 1-n, 4, -4). - Jean-François Alcover, Apr 30 2012

a(n) ~ 2*5^(n+1/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012

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

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

Extensions

More terms from Emeric Deutsch, May 11 2007