A099250 Bisection of Motzkin numbers A001006.

1, 4, 21, 127, 835, 5798, 41835, 310572, 2356779, 18199284, 142547559, 1129760415, 9043402501, 73007772802, 593742784829, 4859761676391, 40002464776083, 330931069469828, 2750016719520991, 22944749046030949, 192137918101841817, 1614282136160911722

Offset

0,2

Comments

a(n) is the number of grand Motzkin paths from (0,0) to (2n+2,0) that avoid vertices (k,0) for all odd k and end on a down step. - Alexander Burstein, May 11 2021

Links

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

Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.

Veronika Irvine, Stephen Melczer, and Frank Ruskey, Vertically constrained Motzkin-like paths inspired by bobbin lace, arXiv:1804.08725 [math.CO], 2018.

Formula

a(n) = (2/Pi)*Integral_{x=-1..1} (1+2*x)^(2*n+1)*sqrt(1-x^2). [Peter Luschny, Sep 11 2011]

Recurrence: (n+1)*(2*n+3)*a(n) = (14*n^2+23*n+6)*a(n-1) + 3*(14*n^2-37*n+21)*a(n-2) - 27*(n-2)*(2*n-3)*a(n-3). - Vaclav Kotesovec, Oct 17 2012

a(n) ~ 3^(2*n+5/2)/(4*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012

a(n) = Sum_{j=0..3*n+4} binomial(j,2*j-5*n-7)*binomial(3*n+4,j) /(3*n+4). [Vladimir Kruchinin, Mar 09 2013]

G.f.: (1/x) * Series_Reversion( x*(1+x) / ( (1+2*x)^2 * (1+x+x^2) ) ). - Paul D. Hanna, Oct 03 2014

Maple

G:=(1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2): GG:=series(G, x=0, 60): seq(coeff(GG, x^(2*n-1)), n=1..24);  # Emeric Deutsch
M := proc(n) option remember; `if`(n<2, 1, (3*(n-1)*M(n-2)+(2*n+1)*M(n-1))/(n+2)) end: A099250 := n -> M(2*n+1):
seq(A099250(i), i=0..20); # Peter Luschny, Sep 11 2011

Mathematica

Take[CoefficientList[Series[(1-x-(1-2x-3x^2)^(1/2))/(2x^2), {x, 0, 60}], x], {2, -1, 2}] (* Harvey P. Dale, Sep 11 2011 *)
Table[Hypergeometric2F1[-1/2-n, -n, 2, 4], {n, 0, 30}] (* Jean-François Alcover, Apr 03 2015 *)
MotzkinNumber = DifferenceRoot[Function[{y, n}, {(-3n-3)*y[n] + (-2n-5)*y[n+1] + (n+4)*y[n+2] == 0, y[0] == 1, y[1] == 1}]];
Table[MotzkinNumber[2n+1], {n, 0, 20}] (* Jean-François Alcover, Oct 27 2021 *)

Prog

(PARI) a(n)=sum(j=0, 3*n+4, binomial(j, 2*j-5*n-7)*binomial(3*n+4, j))/(3*n+4); /* Joerg Arndt, Mar 09 2013 */
(PARI) x='x+O('x^66); v=Vec((1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2)); vector(#v\2, n, v[2*n]) \\ Joerg Arndt, May 12 2013
(PARI) {a(n)=polcoeff(1/x*serreverse( x*(1+x)/((1+2*x)^2*(1+x+x^2) +x^2*O(x^n)) ), n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Oct 03 2014

Crossrefs

Cf. A001006, A026945, A048990.

Sequence in context: A281581 A007345 A255673 * A293192 A300674 A371430

Adjacent sequences: A099247 A099248 A099249 * A099251 A099252 A099253

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 16 2004

Extensions

More terms from Emeric Deutsch, Nov 17 2004

Status

approved