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%I #27 Jul 18 2018 17:50:18
%S 1,1,1,2,3,7,11,28,46,122,207,562,977,2693,4769,13288,23872,67064,
%T 121862,344588,631958,1796518,3319923,9479780,17630692,50532640,
%U 94493713,271710662,510468519,1471935235,2776629563,8026070768,15194389388,44015873308,83591476528
%N Number of Motzkin n-paths avoiding even-numbered steps that are flat steps.
%H Alois P. Heinz, <a href="/A214938/b214938.txt">Table of n, a(n) for n = 0..1000</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, 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) = Sum_{k=0..floor(n/4)} C(floor((n+1)/2), (n mod 2) + 2*(floor(n/4) - k)) * A000108(k + floor((n+2)/4)).
%F Let g.f. A(x) = B(x^2) + x*C(x^2), then
%F B(x) = (1/x)*Series_Reversion( x*(1-x)*(1-2*x)^2 / (1-4*x+5*x^2-2*x^3+x^4) ),
%F C(x) = (1/x)*Series_Reversion( x / (1+2*x+3*x^2+2*x^3 + 2*x^6*Catalan(-x^2)^3) )
%F where Catalan(x) = (1-sqrt(1-4*x))/(2*x). - _Paul D. Hanna_, Aug 03 2012
%F a(n) ~ c * 6^(n/2+1)/(5*sqrt(5*Pi)*n^(3/2)), where c = 2 * sqrt(3) if n is even and c = 3 * sqrt(2) if n is odd. - _Vaclav Kotesovec_, Nov 07 2013
%e a(5) = 7: UuFdD, UuDdF, UdUdF UdFuD, FuUdD, FuFdF, FuDuD, showing even-numbered steps in lower case.
%p a:= proc(n) option remember; `if`(n<7, [1, 1, 1, 2, 3, 7, 11][n+1],
%p (4*(n+1)*(5066415*n^3-39734381*n^2+51596519*n-4935351)*a(n-1)
%p +(83427510*n^4-315565444*n^3-532176102*n^2+1458851596*n
%p +157931232)*a(n-2) -(157058865*n^4-1556016371*n^3
%p +3706209891*n^2+220948511*n-3544991136)*a(n-3) -(107648400*n^4
%p -766240720*n^3+696027720*n^2+4498794592*n -8240373864)*a(n-4)
%p +8*(n-4)*(25332075*n^3-234136810*n^2+385914455*n+722870772)*a(n-5)
%p -24*(n-5)*(1345605*n^3-3657347*n^2-11033479*n+18898695)*a(n-6)
%p +12*(n-5)*(n-6)*(5066415*n^2-14402306*n-21087469)*a(n-7)) /
%p (8*(n+2)*(n+1)*(1345605*n^2-5002952*n-4935351)))
%p end:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Nov 02 2013
%t Table[Sum[Binomial[Floor[(n+1)/2],Mod[n,2]+2*(Floor[n/4]-k)] * CatalanNumber[k+Floor[(n+2)/4]],{k,0,Floor[n/4]}],{n,0,34}]
%o (PARI) /* G.f. A(x) = B(x^2) + x*C(x^2): */ {a(n)=local(A,B,C);
%o B=(1/x)*serreverse(x*(1-x)*(1-2*x)^2/(1-4*x+5*x^2-2*x^3+x^4+x*O(x^n)));
%o C=(1/x)*serreverse(x/(1+2*x+3*x^2+2*x^3+(1-sqrt(1+4*x^2+x*O(x^n)))^3/4));
%o A=subst(B,x,x^2)+x*subst(C,x,x^2); polcoeff(A,n)} \\ _Paul D. Hanna_, Aug 03 2012
%Y Cf. A001006, A000108.
%K nonn
%O 0,4
%A _David Scambler_, Jul 30 2012
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