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A214938 Number of Motzkin n-paths avoiding even-numbered steps that are flat steps. 3
1, 1, 1, 2, 3, 7, 11, 28, 46, 122, 207, 562, 977, 2693, 4769, 13288, 23872, 67064, 121862, 344588, 631958, 1796518, 3319923, 9479780, 17630692, 50532640, 94493713, 271710662, 510468519, 1471935235, 2776629563, 8026070768, 15194389388, 44015873308, 83591476528 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

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

FORMULA

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)).

Let g.f. A(x) = B(x^2) + x*C(x^2), then

B(x) = (1/x)*Series_Reversion( x*(1-x)*(1-2*x)^2 / (1-4*x+5*x^2-2*x^3+x^4) ),

C(x) = (1/x)*Series_Reversion( x / (1+2*x+3*x^2+2*x^3 + 2*x^6*Catalan(-x^2)^3) )

where Catalan(x) = (1-sqrt(1-4*x))/(2*x). - Paul D. Hanna, Aug 03 2012

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

EXAMPLE

a(5) = 7: UuFdD, UuDdF, UdUdF UdFuD, FuUdD, FuFdF, FuDuD, showing even-numbered steps in lower case.

MAPLE

a:= proc(n) option remember; `if`(n<7, [1, 1, 1, 2, 3, 7, 11][n+1],

      (4*(n+1)*(5066415*n^3-39734381*n^2+51596519*n-4935351)*a(n-1)

      +(83427510*n^4-315565444*n^3-532176102*n^2+1458851596*n

       +157931232)*a(n-2) -(157058865*n^4-1556016371*n^3

       +3706209891*n^2+220948511*n-3544991136)*a(n-3) -(107648400*n^4

       -766240720*n^3+696027720*n^2+4498794592*n -8240373864)*a(n-4)

      +8*(n-4)*(25332075*n^3-234136810*n^2+385914455*n+722870772)*a(n-5)

      -24*(n-5)*(1345605*n^3-3657347*n^2-11033479*n+18898695)*a(n-6)

      +12*(n-5)*(n-6)*(5066415*n^2-14402306*n-21087469)*a(n-7)) /

      (8*(n+2)*(n+1)*(1345605*n^2-5002952*n-4935351)))

    end:

seq(a(n), n=0..40);  # Alois P. Heinz, Nov 02 2013

MATHEMATICA

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}]

PROG

(PARI) /* G.f. A(x) = B(x^2) + x*C(x^2): */ {a(n)=local(A, B, C);

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)));

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));

A=subst(B, x, x^2)+x*subst(C, x, x^2); polcoeff(A, n)} \\ Paul D. Hanna, Aug 03 2012

CROSSREFS

Cf. A001006, A000108.

Sequence in context: A305750 A107857 A107858 * A143926 A112840 A014981

Adjacent sequences:  A214935 A214936 A214937 * A214939 A214940 A214941

KEYWORD

nonn

AUTHOR

David Scambler, Jul 30 2012

STATUS

approved

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Last modified August 8 19:29 EDT 2020. Contains 336298 sequences. (Running on oeis4.)