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A104545 Number of Motzkin paths of length n having no consecutive (1,0) steps. 7
1, 1, 1, 3, 5, 11, 25, 55, 129, 303, 721, 1743, 4241, 10415, 25761, 64095, 160385, 403263, 1018369, 2581887, 6569089, 16767871, 42927105, 110194175, 283574017, 731427583, 1890600193, 4896499455, 12704869633, 33021750015, 85966113281 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

a(n) = A104544(n,0) (n > 0).

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..2302

Andrei Asinowski, Cyril Banderier, Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).

Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019.

FORMULA

G.f.: (1-sqrt(1-4z^2*(1+z)^2))/(2z^2*(1+z)).

G.f. A(x) satisfies:

(1) A(x) = (1+x)*(1 + x^2*A(x)^2).

(2) A(x) = exp( Sum_{n>=1} x^n * A(x)^(-n)/n * [Sum_{k=0..n} C(n,k)^2 * x^k * A(x)^(2*k)] ).

(3) A(x) = exp( Sum_{n>=1} x^n * A(x)^(-n)/n * [(1-x/A(x)^2)^(2*n+1) * Sum_{k>=0} C(n+k,k)^2*x^k * A(x)^(2*k)] ).

a(n+1) = sum(binomial(2*k,k)/(k+1)*(binomial(2*k,n-2*k+1)+binomial(2*k,n-2*k)),k,ceiling(n/4),(n+1)/2), a(0)=1. - Vladimir Kruchinin, Mar 14 2012

(n+2)*a(n) + (n+2)*a(n-1) + 4*(-n+1)*a(n-2) + 12*(-n+2)*a(n-3) + 12*(-n+3)*a(n-4) + 4*(-n+4)*a(n-5) = 0. - R. J. Mathar, Jul 23 2017

a(n) ~ 3^(1/4) * (1+sqrt(3))^(n + 1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 17 2017

EXAMPLE

a(3)=3 because we have UDH, HUD and UHD, where U=(1,1), D=(1,-1) and H=(1,0) (HHH does not qualify).

The logarithm of the g.f. A = A(x) equals the series:

log(A(x)) = (1 + x*A^2)*x/A + (1 + 2^2*x*A^2 + x^2*A^4)*x^2/A^2/2 +

(1 + 3^2*x*A^2 + 3^2*x^2*A^4 + x^3*A^6)*x^3/A^3/3 +

(1 + 4^2*x*A^2 + 6^2*x^2*A^4 + 4^2*x^3*A^6 + x^4*A^8)*x^4/A^4/4 +

(1 + 5^2*x*A^2 + 10^2*x^2*A^4 + 10^2*x^3*A^6 + 5^2*x^4*A^8 + x^5*A^10)*x^5/A^5/5 + ...

MAPLE

G:=(1-sqrt(1-4*z^2*(1+z)^2))/2/z^2/(1+z): Gser:=series(G, z=0, 35): 1, seq(coeff(Gser, z^n), n=1..31);

MATHEMATICA

Array[Sum[Binomial[2 k, k]/(k + 1) (Binomial[2 k, # - 2 k + 1] + Binomial[2 k, # - 2 k]), {k, Ceiling[#/4], (# + 1)/2}] &[# - 1] &, 31, 0] (* Michael De Vlieger, Feb 18 2020 *)

CoefficientList[Series[(1-Sqrt[1-4x^2 (1+x)^2])/(2x^2 (1+x)), {x, 0, 30}], x] (* Harvey P. Dale, Mar 02 2020 *)

PROG

(PARI) {a(n)=local(p=-1, q=2, A=1+x); for(i=1, n, A=(1+x*A^(p+1))*(1+x^2*A^(p+q+1))+x*O(x^n)); polcoeff(A, n)}

(PARI) {a(n)=local(p=-1, q=2, A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*(A+x*O(x^n))^(p*m)/m*sum(j=0, m, binomial(m, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}

(PARI) {a(n)=local(p=-1, q=2, A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*(A+x*O(x^n))^(p*m)/m*(1-x*A^q)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}

(Maxima)

b(n):=sum(binomial(2*k, k)/(k+1)*(binomial(2*k, n-2*k+1)+binomial(2*k, n-2*k)), k, ceiling(n/4), (n+1)/2); a(n):=if n=0 then 1 else b(n-1); /* Vladimir Kruchinin, Mar 14 2012 */

CROSSREFS

Cf. A001006, A104544, A198957, A200718.

Sequence in context: A261896 A285184 A018008 * A027050 A240148 A109249

Adjacent sequences:  A104542 A104543 A104544 * A104546 A104547 A104548

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Mar 14 2005

STATUS

approved

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Last modified July 8 03:25 EDT 2020. Contains 335503 sequences. (Running on oeis4.)