A257389 Number of 3-generalized Motzkin paths of length n with no level steps H=(3,0) at odd level.

1, 0, 1, 1, 2, 2, 6, 6, 17, 21, 54, 74, 183, 272, 644, 1025, 2342, 3928, 8734, 15264, 33227, 59989, 128484, 238008, 503563, 952038, 1995955, 3835381, 7987092, 15548654, 32223061, 63388488, 130918071, 259724317, 535168956, 1069025128

Offset

0,5

Links

Robert Israel, Table of n, a(n) for n = 0..3087

Formula

G.f.: (1-x^3-sqrt((1-x^3)*(1-4*x^2-x^3)))/(2*x^2*(1-x^3)).

a(n) = Sum_{k=0..n/3}(((-1)^(n-3*k)+1)*(binomial((n-k)/2,k)*(binomial(n-3*k,(n-3*k)/2))/((n-3*k+2)))). - Vladimir Kruchinin, Apr 02 2016

(2 + n)*a(n) + (14 + 4*n)*a(n + 1) + (-10 - 2*n)*a(n + 3) + (-20 - 4*n)*a(n + 4) + (8 + n)*a(n + 6) = 0. - Robert Israel, Nov 04 2019

Example

For n=6 we have 6 paths: UDUDUD, H3H3, UUDUDD, UUUDDD, UDUUDD and UUDDUD, where H3=(3,0).

Maple

f:= gfun:-rectoproc({(2 + n)*a(n) + (14 + 4*n)*a(n + 1) + (-10 - 2*n)*a(n + 3) + (-20 - 4*n)*a(n + 4) + (8 + n)*a(n + 6), a(0) = 1, a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 2, a(5) = 2}, a(n), remember):
map(f, [$0..100]); # Robert Israel, Nov 04 2019

Prog

(Maxima)
a(n):=sum(((-1)^(n-3*k)+1)*((binomial((n-k)/2, k) )*(binomial(n-3*k, (n-3*k)/2))/((n-3*k+2))), k, 0, (n)/3); /* Vladimir Kruchinin, Apr 02 2016 */

Crossrefs

Cf. A000108, A090344, A086622, A257178, A007317, A064613, A104455, A104498.

Sequence in context: A287603 A034422 A140833 * A071908 A370588 A011260

Adjacent sequences: A257386 A257387 A257388 * A257390 A257391 A257392

Keyword

nonn

Author

José Luis Ramírez Ramírez, Apr 21 2015

Status

approved