A086871 Row sums of A059450.

1, 2, 10, 58, 370, 2514, 17850, 130890, 983650, 7536418, 58648810, 462306266, 3683602130, 29620138994, 240059315610, 1958940281322, 16081662931650, 132723191430210, 1100568370427850, 9164925012016506, 76612776253995570

Offset

0,2

Comments

Hankel transform is A165928. - Paul Barry, Sep 30 2009

Number of skew Dyck paths of semilength n with the down steps coming in two colors. - David Scambler, Jun 21 2013

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

C. Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003), 13-28.

J. Machacek, Lattice walks ending on a coordinate hyperlane avoiding backtracking and repeats, arXiv:2105.02417 [math.CO], 2021. See Thm. 4.4 G(x,E^1).

Formula

a(n) = 2*A059231(n), if n>0.

G.f.: (1 - x - sqrt((1 - x) * (1 - 9*x))) / (4*x) = 2 / (1 + sqrt((1 - 9*x) / (1 - x))) =: y satisfies 0 = (1 - x) * (1 - y) + 2*x*y^2. - Michael Somos, Mar 06 2004

Moment representation: a(n) = (1/(4*Pi))*Integral_{x=1..9} x^n*sqrt(-x^2+10x-9)/x+(1/2)*0^n. - Paul Barry, Sep 30 2009

D-finite with recurrence Recurrence: (n+1)*a(n) = 5*(2*n-1)*a(n-1) - 9*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 14 2012

a(n) ~ 3^(2*n+1)/(2*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012

G.f.: 1/(2*x) - 1/2 + G(0) where G(k) = 1 - 1/(x + x/(1 + 1/G(k+1) )) ; (continued fraction,3-step). - Sergei N. Gladkovskii, Nov 29 2012

Example

G.f. = 1 + 2*x + 10*x^2 + 58*x^3 + 370*x^4 + 2514*x^5 + 17850*x^6 + 130890*x^7 + ...

Mathematica

Table[SeriesCoefficient[2/(1+Sqrt[(1-9*x)/(1-x)]), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( 2 / (1 + sqrt((1 - 9*x) / (1 - x) + x * O(x^n))), n))}; /* Michael Somos, Mar 06 2004 */
(PARI) {a(n) = if( n<1, n==0, n++; 2 * polcoeff( serreverse( x * (1 - 4*x) / (1 - 3*x) + x * O(x^n)), n))}; /* Michael Somos, Mar 06 2004 */

Crossrefs

Sequence in context: A278095 A075870 A074608 * A108450 A293111 A112369

Adjacent sequences: A086868 A086869 A086870 * A086872 A086873 A086874

Keyword

nonn,easy

Author

N. J. A. Sloane, Sep 16 2003

Extensions

More terms from Ray Chandler, Sep 17 2003

Status

approved