A050253 G.f.: ( 1 - x^2 - sqrt( 1 - 2*x^2 - 4*x^3 - 3*x^4 ) ) / ( 2*x^3 ).

1, 1, 1, 2, 3, 5, 9, 16, 29, 54, 101, 191, 365, 702, 1359, 2647, 5181, 10187, 20113, 39856, 79243, 158036, 316053, 633689, 1273559, 2565136, 5177043, 10468199, 21204379, 43022215, 87423573, 177906552, 362531425, 739700055, 1511091377

Offset

0,4

Comments

a(n)=number of Motzkin (n-1)-paths (A001006) containing no three consecutive weakly-rising steps (n>=1). A weakly-rising step is an upstep or flatstep. For example, a(5)=5 counts FUDF, UDFF, UDUD, UFDF, UUDD while the path FUFD, say, is not counted because the first 3 steps are weakly-rising. - David Callan, Oct 25 2004

Hankel transform is A010892(n+1). - Paul Barry, Jul 29 2010

Links

Robert Israel, Table of n, a(n) for n = 0..3056

Formula

a(n) = A108296(n+2) - A108296(n). - Paul Barry, May 31 2005

G.f.: 1/(1-x-x^3/(1-x^2-x^3/(1-x-x^3/(1-x^2-x^3/(1-x-x^3/(1-... (continued fraction). - Paul Barry, May 25 2009

G.f.: 1/(1-x/(1-x^2/(1-x^3/(1-x/(1-x^2/(1-x^3/(1-x/(1-x^2/(1-x^3/(1-... (continued fraction). - Paul Barry, Jul 29 2010

D-finite with recurrence: (n+3)*a(n) + (n+2)*a(n-1) - 2n*a(n-2) + 2*(4-3n)*a(n-3) + (19-7n)*a(n-4) + 3*(4-n)*a(n-5) = 0. - R. J. Mathar, Nov 15 2011

From Robert Israel, Jan 15 2018: (Start)

Recurrence verified using the differential equation (3*x^5+4*x^4+2*x^3-x)*y' + (3*x^4+6*x^3+4*x^2-3)*y + x^2+4*x+3 = 0 satisfied by the g.f.

(3+3*n)*a(n) + (10+4*n)*a(1+n) + (2*n+8)*a(n+2) + (-7-n)*a(n+4) = 0. (End)

a(n) = Sum_{k=1..n} ((Sum_{j=0..k} C(j,n-k-j)*C(k,j)))*C(n-k,k-1))/k). - Vladimir Kruchinin, Nov 21 2014

G.f. A(x) satisfies A(x) = x*(1+sqrt(1+4*(A(x)+A(x)^2+A(x)^3))/2. - Vladimir Kruchinin, Nov 21 2014

a(0) = a(1) = 1; a(n) = a(n-2) + Sum_{k=0..n-3} a(k) * a(n-k-3). - Ilya Gutkovskiy, Apr 11 2021

a(n) ~ sqrt(6 + 3*r - 6*r^2) * (6 + 5*r + 6*r^2) * (1 + r + 3*r^2)^n / (6*sqrt(Pi)*n^(3/2)), where r = 0.4693964245699946792019209673920017843813793... is the root of the equation 3*r^3 + r^2 + r - 1 = 0. - Vaclav Kotesovec, Jul 03 2021

Maple

f:= gfun:-rectoproc({(3+3*n)*a(n)+(10+4*n)*a(1+n)+(2*n+8)*a(n+2)+(-7-n)*a(n+4), a(0) = 1, a(1) = 1, a(2) = 1, a(3) = 2}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Jan 15 2018

Mathematica

CoefficientList[Series[(1-x^2-Sqrt[1-2x^2-4x^3-3x^4])/(2x^3), {x, 0, 40}], x] (* Harvey P. Dale, Jul 17 2015 *)

Prog

(Maxima) a(n):=if n=0 then 1 else sum(((sum(binomial(j, n-k-j)*binomial(k, j), j, 0, k))*binomial(n-k, k-1))/k, k, 1, n); /* Vladimir Kruchinin, Nov 21 2014 */

Crossrefs

Sequence in context: A103285 A000049 A000050 * A198518 A182558 A298204

Adjacent sequences: A050250 A050251 A050252 * A050254 A050255 A050256

Keyword

easy,nonn

Author

Emanuele Munarini, May 09 2003

Status

approved