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A257389 Number of 3-generalized Motzkin paths of length n with no level steps H=(3,0) at odd level. 2
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 (list; graph; refs; listen; history; text; internal format)
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 A011260 A117855

Adjacent sequences:  A257386 A257387 A257388 * A257390 A257391 A257392

KEYWORD

nonn,changed

AUTHOR

José Luis Ramírez Ramírez, Apr 21 2015

STATUS

approved

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Last modified November 17 06:06 EST 2019. Contains 329217 sequences. (Running on oeis4.)