%I #70 Jan 11 2025 03:28:00

%S 1,4,21,127,835,5798,41835,310572,2356779,18199284,142547559,

%T 1129760415,9043402501,73007772802,593742784829,4859761676391,

%U 40002464776083,330931069469828,2750016719520991,22944749046030949,192137918101841817,1614282136160911722

%N Bisection of Motzkin numbers A001006.

%C 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

%H Vincenzo Librandi, <a href="/A099250/b099250.txt">Table of n, a(n) for n = 0..200</a>

%H Veronika Irvine, <a href="http://hdl.handle.net/1828/7495">Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns</a>, PhD Dissertation, University of Victoria, 2016.

%H Veronika Irvine, Stephen Melczer, and Frank Ruskey, <a href="https://arxiv.org/abs/1804.08725">Vertically constrained Motzkin-like paths inspired by bobbin lace</a>, arXiv:1804.08725 [math.CO], 2018.

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

%F 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

%F a(n) ~ 3^(2*n+5/2)/(4*sqrt(2*Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 17 2012

%F G.f.: (1/x) * Series_Reversion( x*(1+x) / ( (1+2*x)^2 * (1+x+x^2) ) ). - _Paul D. Hanna_, Oct 03 2014

%F From _Peter Bala_, Apr 20 2024: (Start)

%F a(n) = Sum_{k = 0..2*n+1} (-1)^(k+1) * binomial(2*n+1, k)*Catalan(k+1).

%F a(n) = Sum_{k = 0..2*n+1} (-1)^k * binomial(2*n+1, k)*Catalan(k+1)*3^(2*n-k+1).

%F (4*n - 1)*(2*n + 3)*(n + 1)*a(n) = 2*(4*n + 1)*(10*n^2 + 5*n - 3)*a(n-1) - 9*(4*n + 3)*(2*n - 1)*(n - 1)*a(n-2) with a(0) = 1 and a(1) = 4. (End)

%p 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_

%p 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):

%p seq(A099250(i),i=0..20); # _Peter Luschny_, Sep 11 2011

%t 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 *)

%t Table[Hypergeometric2F1[-1/2-n, -n, 2, 4], {n, 0, 30}] (* _Jean-François Alcover_, Apr 03 2015 *)

%t 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}]];

%t Table[MotzkinNumber[2n+1], {n, 0, 20}] (* _Jean-François Alcover_, Oct 27 2021 *)

%o (PARI) my(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

%o (PARI) {a(n)=polcoeff(1/x*serreverse( x*(1+x)/((1+2*x)^2*(1+x+x^2) +x^2*O(x^n)) ),n)}

%o for(n=0,30,print1(a(n),", ")) \\ _Paul D. Hanna_, Oct 03 2014

%Y Cf. A001006, A026945, A048990.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Nov 16 2004

%E More terms from _Emeric Deutsch_, Nov 17 2004