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A086871 Row sums of A059450. 3
1, 2, 10, 58, 370, 2514, 17850, 130890, 983650, 7536418, 58648810, 462306266, 3683602130, 29620138994, 240059315610, 1958940281322, 16081662931650, 132723191430210, 1100568370427850, 9164925012016506, 76612776253995570 (list; graph; refs; listen; history; text; internal format)
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

REFERENCES

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

LINKS

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

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))*Int(x^n*sqrt(-x^2+10x-9)/x,x,1,9)+(1/2)*0^n - Paul Barry, Sep 30 2009

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

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Last modified August 25 09:38 EDT 2019. Contains 326323 sequences. (Running on oeis4.)